# On $L^{1}$ estimates of solutions
of compressible viscoelastic system

Yusuke Ishigaki

(Department of Mathematics,
Tokyo Institute of Technology,
Tokyo 152-8551, Japan,
<EMAIL_ADDRESS>
)

###### Abstract

We consider the large time behavior of solutions of compressible viscoelastic
system around a motionless state in a three-dimensional whole space. We show
that if the initial preturbation belongs to $W^{2,1}$, and is sufficiently
small in $H^{4}\cap L^{1}$, the solutions grow in time at the same rate as
$t^{\frac{1}{2}}$ in $L^{1}$ due to diffusion wave phenomena of the system
caused by interaction between sound wave, viscous diffusion and elastic wave.

Keywords: Compressible viscoelastic system; diffusion wave; large time
behavior.

##  1 Introduction

This paper studies the initial value problem of compressible viscoelastic
system

$\displaystyle\partial_{t}\rho+\mathrm{div}(\rho v)=0,$ (1.1)
$\displaystyle\rho(\partial_{t}v+v\cdot\nabla
v)-\nu{\Delta}v-(\nu+\nu^{\prime})\nabla\mathrm{div}v+\nabla
P(\rho)=\beta^{2}\mathrm{div}(\rho F{}^{\top}\\!F),$ (1.2)
$\displaystyle\partial_{t}F+v\cdot\nabla F=(\nabla v)F,$ (1.3)
$\displaystyle(\rho,v,F)|_{t=0}=(\rho_{0},v_{0},F_{0}).$ (1.4)

in the whole space $\mathbb{R}^{3}$. Here $\rho=\rho(x,t)$,
$v={}^{\top}(v^{1}(x,t),v^{2}(x,t),v^{3}(x,t))$, and $F=(F^{jk}(x,t))_{1\leq
j,k\leq 3}$ denote the unknown density, velocity field, and deformation
tensor, respectively, at position $x\in\mathbb{R}^{3}$ and time $t\geq 0$;
$P=P(\rho)$ is the given pressure; $\nu$ and $\nu^{\prime}$ are the viscosity
coefficients satisfying

$\nu>0,~{}2\nu+3\nu^{\prime}\geq 0;$

$\beta>0$ is the propagation speed of elastic wave. We assume that
$P^{\prime}(1)>0$, and set $\gamma=\sqrt{P^{\prime}(1)}$.

We also impose the following conditions for $\rho_{0}$ and $F_{0}$

$\displaystyle\rho_{0}\mathrm{det}F_{0}=1,$ (1.5)
$\displaystyle\sum_{m=1}^{3}(F_{0}^{ml}\partial_{x_{m}}F_{0}^{jk}-F_{0}^{mk}\partial_{x_{m}}F_{0}^{jl})=0,~{}j,k,l=1,2,3,$
(1.6) $\displaystyle\mathrm{div}(\rho_{0}{}^{\top}\\!F_{0})=0.$ (1.7)

It follows from [5, Appendix A] and [18, Proposition.1] that the quantities
$(1.5)$ and $(1.6)$ are invariant for $t\geq 0$:

$\displaystyle\rho\mathrm{det}F=1,$ (1.8)
$\displaystyle\sum_{m=1}^{3}(F^{ml}\partial_{x_{m}}F^{jk}-F^{mk}\partial_{x_{m}}F^{jl})=0,~{}j,k,l=1,2,3.$
(1.9)

Here the constraint $(1.8)$ means compressibility of fluid and the constraint
$(1.9)$ called the Piola’s formula is derived from a certain symmetric
property of the first order derivatives of $F$ in the Lagrangian coordinates.
Furthermore we see from [7. Appendix A] and [8. Appendix A] that the
constraints $(1.8)$ and $(1.9)$ lead to the time invariance of the quantity
$(1.7)$:

$\displaystyle\mathrm{div}(\rho{}^{\top}\\!F)=0.$ (1.10)

The purpose of this paper is to deduce the estimate of the $L^{1}$ norm of
solutions of the problem $(1.1)$–$(1.7)$ around a motionless state
$(\rho,v,F)=(1,0,I)$. Here $I$ is the $3\times 3$ identity matrix.

The system $(1.1)$–$(1.3)$ is derived from a motion of compressible
viscoelastic fluid in the macroscopic scale under the Hookean li3near
elasticity by the variational settings. We refer to [1, 13, 21] for more
physical details. We can classify the system $(1.1)$–$(1.3)$ in a quasilinear
parabolic-hyperbolic system since the system $(1.1)$–$(1.3)$ consists of the
compressible Navier-Stokes equations and a first order hyperbolic system for
$F$.

In the case $\beta=0$, the large time behavior of the solutions around
$(\rho,v)=(1,0)$ has been investigated so far. In particular, if we set
$\beta=0$ formally, the system $(1.1)$–$(1.3)$ becomes the usual compressible
Navier-Stokes equation

$\displaystyle\partial_{t}\rho+\mathrm{div}(\rho v)=0,$ (1.11)
$\displaystyle\rho(\partial_{t}v+v\cdot\nabla
v)-\nu{\Delta}v-(\nu+\nu^{\prime})\nabla\mathrm{div}v+\nabla P(\rho)=0,$
(1.12) $\displaystyle(\rho,v)|_{t=0}=(\rho_{0},v_{0}).$ (1.13)

Matsumura and Nishida [15] showed the global existence of the solutions of the
problem $(1.11)$–$(1.13)$ provided that the initial perturbation is
sufficiently small in $H^{3}\cap L^{1}$, and derived the decay estimate:

$\|\nabla^{k}(\phi(t),m(t))\|_{L^{2}}\leq
C(1+t)^{-\frac{3}{4}-\frac{k}{2}},~{}k=0,1,$

where $(\phi,m)=(\rho-1,\rho v)$. Hoff and Zumbrun [2] derived the following
$L^{p}$ decay estimates and asymptotic properties in $\mathbb{R}^{n},~{}n\geq
2$:

$\displaystyle\|(\phi(t),m(t))\|_{L^{p}}\leq\begin{cases}C(1+t)^{-\frac{n}{2}\left(1-\frac{1}{p}\right)-\frac{n-1}{4}\left(1-\frac{2}{p}\right)}L(t),&1\leq
p<2,\\\ C(1+t)^{-\frac{n}{2}\left(1-\frac{1}{p}\right)},&2\leq
p\leq\infty,\end{cases}$
$\displaystyle\begin{array}[]{l}\left\|\left((\phi(t),m(t))-\left(0,\mathcal{F}^{-1}\left(e^{-\nu|\xi|^{2}t}\hat{\mathcal{P}}(\xi)\hat{m}_{0}\right)\right)\right)\right\|_{L^{p}}\\\\[6.45831pt]
\hskip 6.45831pt\leq
C(1+t)^{-\frac{n}{2}\left(1-\frac{1}{p}\right)-\frac{n-1}{4}\left(1-\frac{2}{p}\right)},\end{array}~{}2\leq
p\leq\infty,$

provided that the initial perturbation is sufficiently small in $H^{4}\cap
L^{1}$, where $L(t)=\log(1+t)$ when $n=2$, and $L(t)=1$ when $n\geq 3$. Here
$\hat{\mathcal{P}}(\xi)=I-\frac{{\xi}^{\top}\xi}{|\xi|^{2}},~{}\xi\in\mathbb{R}^{n}$.
According to [10], the solution of the linearized system is decomposed as the
sum of two terms, one is the incompressible part given by
$\mathcal{F}^{-1}\left(e^{-\nu|\xi|^{2}t}\hat{\mathcal{P}}(\xi)\hat{m}_{0}\right)$
which behaves pure diffusively, and the other is the compressible part
$(\phi(t),m(t))-\left(0,\mathcal{F}^{-1}\left(e^{-\nu|\xi|^{2}t}\hat{\mathcal{P}}(\xi)\hat{m}_{0}\right)\right)$
containing the diffusion wave which stands for the convolution of the heat
kernel and the fundamental solution of the wave equation with sound speed
$\gamma$. The authors of [2] revealed the hyperbolic aspect of the system
$(1.11)$–$(1.12)$ by proving that the large time behavior of the compressible
part is different from the heat kernel in $L^{p}$ as $t\to\infty$, except the
case $p=2$. See also [11] for the linearized problem.

In the case $\beta>0$, the mathematical analysis of solutions of the initial
value problem $(1.1)$–$(1.7)$ around the motionless state have been developed
so far. The local existence of the strong solution of the problem
$(1.1)$–$(1.7)$ was guaranteed by Hu and Wang [4]. The global existence of the
strong solution of the initial value problem $(1.1)$–$(1.7)$ was proved by Hu
and Wang [5], Qian and Zhang [18], and Hu and Wu [6], provided that the
initial perturbation $(\rho_{0}-1,v_{0},F_{0}-I)$ is sufficiently small. Hu
and Wu [6] showed that if the initial perturbation
$(\rho_{0}-1,v_{0},F_{0}-I)$ belongs to $L^{1}\cap H^{3}$, the $L^{p}$ decay
estimates hold for the case $2\leq p\leq 6$:

$\displaystyle\|u(t)\|_{L^{p}}\leq
C(1+t)^{-\frac{3}{2}\left(1-\frac{1}{p}\right)},$ (1.14)

by using the Fourier splitting method and the Hodge decomposition. Here
$u(t)=(\phi,w,G)=(\rho-1,v,F-I)$. Moreover, the authors also derived the lower
$L^{2}$ estimate

$\displaystyle\|u(t)\|_{L^{2}}\geq c(1+t)^{-\frac{3}{4}},~{}t\gg 1,$ (1.15)

provided that the following conditions satisfy in the low frequency part
$|\xi|\ll 1$:

$\displaystyle|\hat{\phi}_{0}(\xi)|\geq
c_{0},|\hat{m}_{0}(\xi)|+|\hat{\mathcal{G}}_{0}(\xi)-{}^{\top}\hat{\mathcal{G}}_{0}(\xi)|\ll|\xi|^{\eta_{0}},$
(1.16)

where
$(\phi_{0},m_{0},\mathcal{G}_{0})=(\rho_{0}-1,\rho_{0}v_{0},\rho_{0}F_{0}-I)$;
$c$, $c_{0}$ and $\eta_{0}$ are positive constants independent of $\xi$ and
$t$. Li, Wei and Yao [12, 22] generalized the upper $L^{p}$ decay estimates
$(1.14)$ to the case $2\leq p\leq\infty$, and obtained the $L^{2}$ decay
estimates of higher order derivatives:

$\displaystyle\|\nabla^{k}u(t)\|_{L^{2}}\leq
C(1+t)^{-\frac{3}{4}-\frac{k}{2}},~{}k=0,1,\ldots,N-1,$ (1.17)

provided that $u_{0}=(\rho_{0}-1,v_{0},F_{0}-I)$ belongs to $H^{N},~{}N\geq
3$, and is small in $L^{1}\cap H^{3}$. We also refer to [3, 14, 23] in recent
progresses.

One problem which interests us is that the decay rates in $(1.14)$ reveal only
the parabolic aspect of the system $(1.1)$–$(1.3)$; it would be desirable to
establish decay estimates which reflect the hyperbolic aspect of the system
$(1.1)$–$(1.3)$.

In view of the results in [2], it is expected that the system $(1.1)$–$(1.3)$
has the diffusion wave phenomena affected by sound wave, viscous diffusion and
elastic wave. For simplicity, let us consider the linearized system around
$(1,0,I)$:

$\displaystyle\partial_{t}u+Lu=0.$ (1.18)

Here $L$ denotes the linearized operator given by

$L=\left(\begin{array}[]{@{\ }cc@{\ }cc@{\ } }0&\mathrm{div}&0\\\
\gamma^{2}\nabla&-\nu\Delta-\tilde{\nu}\nabla\mathrm{div}&-\beta^{2}\mathrm{div}\\\
0&-\nabla&0\end{array}\right),$

where $\tilde{\nu}=\nu+\nu^{\prime}$. We then see that the solenoidal part of
the velocity $w_{s}=\mathcal{F}^{-1}(\hat{\mathcal{P}}(\xi)\hat{w})$ satisfies
the following linear symmetric parabolic-hyperbolic system:

$\left\\{\begin{array}[]{l}\partial_{t}w_{s}-\nu\Delta
w_{s}-\beta\mathrm{div}\tilde{G}_{s}=0,\\\
\partial_{t}\tilde{G}_{s}-\beta\nabla w_{s}=0,\end{array}\right.$

where $\tilde{G}_{s}=\beta\mathcal{F}^{-1}(\hat{\mathcal{P}}(\xi)\hat{G})$,
while the complimentary part $w_{c}=w-w_{s}$ solves the following strongly
damped wave equation:

$\partial_{t}^{2}w_{c}-(\beta^{2}+\gamma^{2})\Delta
w_{c}-(\nu+\tilde{\nu})\partial_{t}\Delta w_{c}=0.$

Owing to the results in [20], the large time behavior of the solution of
$(1.18)$ becomes different to the case $\beta=0$ ([2,11]) since the additional
hyperbolic aspect arises in the incompressible part due to elastic wave. As a
result, the principal part of the linearized system $(1.18)$ can be identified
as a system of the strongly damped wave equation. In [9], the hyperbolic
aspect of the system $(1.1)$–$(1.3)$ is clarified by showing the following
$L^{p}$ decay estimates:

$\displaystyle\|u(t)\|_{L^{p}}\leq
C(1+t)^{-\frac{3}{2}\left(1-\frac{1}{p}\right)-\frac{1}{2}\left(1-\frac{2}{p}\right)}.$
(1.19)

for the case $1<p\leq\infty$, provided that the initial perturbation
$(\rho_{0}-1,v_{0},F_{0}-I)$ is small in $L^{1}\cap H^{3}$, This improves the
results in [6, 12].

The main difficulty of the mathematical analysis is nonlinearity of the
constraints $(1.8)$–$(1.10)$. Therefore, straightforward application of the
semigroup theory to the nonlinear problem does not seem valid. To bypass this
difficulty, Hu and Wu [6] found that the behavior of $G$ is controlled by its
skew-symmetric part $G-{}^{\top}G$ due to the constraints $(1.9)$ and
$(1.10)$. This property leads to the global in time existence theorem. The
authors of [6] next used the Helmholtz decomposition of $w$ and the skew-
symmetric part of $G$ to obtain $(1.14)$ with $2\leq p\leq 6$, $(1.17)$ with
$N=2$ and $(1.15)$. In [9], the author relied on a material coordinate
transform which makes the constraint $(1.10)$ a linear one to apply the
analysis of the linearized problem to the nonlinear problem. Let us introduce
a displacement vector $\tilde{\psi}=x-X$ as in [19, 21]:

$\tilde{\psi}(x,t)=x-X(x,t).$

Here $x=x(X,t)$ is the material coordinate defined under the flow map

$\left\\{\begin{array}[]{l}\displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}=v(x(X,t),t),\\\\[4.30554pt]
x(X,0)=X,\end{array}\right.$

and $X=X(x,t)$ denotes the inverse of $x$. In the continuum mechanics theory,
$F$ is given by the Jacobi matrix of $x$ in the material coordinate. Then we
notice that $F$ is written as

$\displaystyle F-I=\nabla\tilde{\psi}+h(\nabla\tilde{\psi}).$ (1.20)

Here $h(\nabla\tilde{\psi})$ is a function satisfying
$h(\nabla\tilde{\psi})=O(|\nabla\tilde{\psi}|^{2})$ for
$|\nabla\tilde{\psi}|\ll 1$. The author next set the nonlinear transform

$\displaystyle\psi=\tilde{\psi}-(-\Delta)^{-1}\mathrm{div}{}^{\top}(\phi\nabla\tilde{\psi}+(1+\phi)h(\nabla\tilde{\psi})).$
(1.21)

Here $(-\Delta)^{-1}=\mathcal{F}^{-1}|\xi|^{-2}\mathcal{F}$. By straight
computation and the constraint $(1.10)$, we arrive at the linear condition
$\phi+\mathrm{tr}(\nabla\psi)=\phi+\mathrm{div}\psi=0$ which makes the
semigroup $e^{-tL}$ generated by $-L$ tend to $0$ as $t\to\infty$ in
$L^{p},~{}p>\frac{5}{4}$. Furthermore, the decay estimate of the
$L^{p}~{}(1<p\leq\infty)$ norm of $u=(\phi,w,G)$ is obtained from
$\tilde{U}=(\phi,w,\nabla\psi)$. Consequently, the $L^{p}$ decay estimates of
$u$ are obtained from the following integral equation

$\tilde{U}(t)=e^{-tL}\tilde{U}(0)+\int_{0}^{t}e^{-(t-s)L}N(\tilde{U}(s))\mathrm{d}s,$

where $N(\tilde{U})=(N_{1}(\tilde{U}),N_{2}(\tilde{U}),N_{3}(\tilde{U}))$ is a
nonlinearity satisfying $N_{1}(\tilde{U})+\mathrm{tr}N_{3}(\tilde{U})=0$.

The another difficulty arises from the nonlinear transform $(1.21)$ containing
the nonlocal operator $(-\Delta)^{-1}$. We note that the operator
$\nabla(-\Delta)^{-1}\mathrm{div}=\mathcal{F}^{-1}\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\mathcal{F}$
is not bounded from $L^{p}$ into $L^{p}$ if $p=1,\infty$ due to the Riesz
operator. In case $p=\infty$, the above difficulty is avoided by using the
Sobolev inequality and the Plancherel theorem, while the case $p=1$, it is
expected that the solution of $L^{1}$ norm grows as $t\to\infty$ due to the
diffusion phenomena, however, it remains open.

In this paper, we find a different approach of the reformulation to show the
following $L^{1}$ estimate of $u$

$\displaystyle\|(\rho(t)-1,v(t),F(t)-I)\|_{L^{1}}\leq
C(1+t)^{\frac{1}{2}},~{}t>0,$ (1.22)

provided that the initial perturbation $u_{0}=(\rho_{0}-1,v_{0},F_{0}-I)$
belongs to $W^{2,1}$, and is sufficiently small in $H^{4}\cap L^{1}$. We also
prove that if $(\rho_{0},v_{0},F_{0})$ satisfies $(1.16)$, then the following
lower $L^{1}$ estimate holds

$\displaystyle\|(\rho(t)-1,v(t),F(t)-I)\|_{L^{1}}\geq
C(1+t)^{\frac{1}{2}},~{}t\gg 1,$ (1.23)

This indicates that the obtained rate $(1+t)^{\frac{1}{2}}$ in $(1.22)$ is
sharp.

We give an outline of the proof of the main result. We first notice that the
constraint $(1.8)$ is read as $\rho=\mathrm{det}F^{-1}$. Then, by employing
$(1.20)$, we have

$\displaystyle\phi=-\mathrm{div}\tilde{\psi}+O(|\nabla\tilde{\psi}|^{2}),~{}\|\nabla\tilde{\psi}\|_{C(0,\infty;L^{\infty})}\ll
1.$ (1.24)

Therefore, the behavior of $\phi$ can be handled by
$-\mathrm{div}\tilde{\psi}$ under the small perturbation. For simplicity, we
omit the tilde $\tilde{\cdot}$ of $\tilde{\psi}$ here.

We next consider the nonlinear problem for
$U=(\tilde{\phi},w,\tilde{G})=(-\mathrm{div}\psi,w,\nabla\psi)$:

$\left\\{\begin{array}[]{l}\partial_{t}U+LU=N(U),\\\
\tilde{\phi}+\mathrm{tr}\tilde{G}=0,~{}\tilde{G}=\nabla\psi,\\\
U|_{t=0}=U_{0}.\end{array}\right.$ (1.25)

where $N(U)=(N_{1}(U),N_{2}(U),N_{3}(U))$ is a nonlinearity such that
$N_{1}(U)+\mathrm{tr}N_{3}(U)=0$. We see from $(1.20)$ and $(1.24)$ that the
$L^{1}$ norm of $u=(\phi,w,G)$ is estimated by
$U=(-\mathrm{div}\psi,w,\nabla\psi)$. We point out that since $U$ and $N(U)$
hold the same linear constraint as in [9], the linear semigroup $e^{-tL}U_{0}$
and the Duammel term $\int_{0}^{t}e^{-(t-s)L}N(U(s))\mathrm{d}s$ do not
include terms which are time-independent or unbounded in $L^{1}$.
Consequently, the $L^{1}$ estimate $(1.22)$ is obtained from the following
integral equation of $U$

$U(t)=e^{-tL}U(0)+\int_{0}^{t}e^{-(t-s)L}N(U(s))\mathrm{d}s.$

The lower $L^{1}$ estimate $(1.23)$ is obtained by the lower $L^{2}$ estimate
$(1.15)$, the estimate $(1.19)$ with $p=\infty$, and the interpolation
inequality.

We mention that this proof does not need the non-local operator in this
reformation. Therefore we remove the difficulty in the case $p=1$, and
simplify the analysis of the solutions around the motionless state.

This paper is organized as follows. In Section 2 we introduce some notations
and function spaces. In Section 3 we state the main result of this paper on
the $L^{1}$ estimates of the solution. In Section 4 we show the $L^{1}$
estimates. In Section 5, we derive the $L^{1}$ estimate of the high frequency
part of the linear semigroup.

##  2 Notation

In this section, we prepare notations and function spaces which will be used
throughout the paper. $L^{p}~{}(1\leq p\leq\infty)$ denotes the usual Lebesgue
space on $\mathbb{R}^{3}$, and its norm is denoted by $\|\cdot\|_{L^{p}}$.
Similarly $W^{m,p}(1\leq p\leq\infty,m\in\\{0\\}\cup\mathbb{N})$ denotes the
$m$-th order $L^{p}$ Sobolev space on $\mathbb{R}^{3}$, and its norm is
denoted by $\|\cdot\|_{W^{m,p}}$. We define $H^{m}=W^{m,2}$ for an integer
$m\geq 0$. For simplicity, we denote
$L^{p}=L^{p}\times(L^{p})^{3}\times(L^{p})^{9}$ (resp.
$H^{m}=H^{m}\times(H^{m})^{3}\times(H^{m})^{9}$).

The inner product of $L^{2}$ is denoted by

$(f,g):=\int_{\mathbb{R}^{3}}f(x)\overline{g(x)}dx,~{}f,g\in L^{2}.$

Here the symbol $\overline{\cdot}$ stands for its complex conjugate. Partial
derivatives of a function $u$ in $x_{j}~{}(j=1,2,3)$ and $t$ are denoted by
$\partial_{x_{j}}u$ and $\partial_{t}u$, respectively. $\Delta$ denotes the
usual Laplacian with respect to $x$. For a multiindex
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in(\\{0\\}\cup\mathbb{N})^{3}$ and
$\xi={}^{\top}(\xi_{1},\xi_{2},\xi_{3})\in\mathbb{R}^{3}$, we define
$\partial_{x}^{\alpha}$ and $\xi^{\alpha}$ as
$\partial_{x}^{\alpha}=\partial_{x_{1}}^{\alpha_{1}}\partial_{x_{2}}^{\alpha_{2}}\partial_{x_{3}}^{\alpha_{3}}$
and
$\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\xi_{2}^{\alpha_{2}}\xi_{3}^{\alpha_{3}}$,
respectively. For a function $u$ and a nonnegative integer $k$, $\nabla^{k}u$
stands for $\nabla^{k}u=\\{\partial_{x}^{\alpha}u|~{}|\alpha|=k\\}$.

For a scalar valued function $\rho=\rho(x)$, we denote by $\nabla\rho$ its
gradient with respect to $x$. For a vector valued function
$w=w(x)={}^{\top}(w^{1}(x),w^{2}(x),w^{3}(x))$, we denote by $\mathrm{div}w$
and $(\nabla w)^{jk}=(\partial_{x_{k}}w^{j})$ its divergence and Jacobian
matrix with respect to $x$, respectively. For a $3\times 3$-matrix valued
function $F=F(x)=(F^{jk}(x))$, we define its divergence $\mathrm{div}F$, trace
$\mathrm{tr}F$ and determinant $\mathrm{det}F$ by
$(\mathrm{div}F)^{j}=\sum_{k=1}^{3}\partial_{x_{k}}F^{jk}$,
$\mathrm{tr}F=\sum_{k=1}^{3}F^{kk}$ and $\mathrm{det}F=\sum_{\sigma\in
S_{3}}\mathrm{sgn}(\sigma)F^{1\sigma(1)}F^{2\sigma(2)}F^{3\sigma(3)}$,
respectively. Here $S_{3}$ denotes a third-order symmetric group; for a
permutation $\sigma\in S_{3}$, we denote by $\mathrm{sgn}(\sigma)$ its
signature.

For functions $f=f(x)$ and $g=g(x)$, we denote by $f*g$ the convolution of $f$
and $g$ :

$(f*g)(x)=\int_{\mathbb{R}^{3}}f(x-y)g(y)\mathrm{d}y.$

We denote by $\hat{f}$ or $\mathcal{F}f$ the Fourier transform of a function
$f=f(x)$:

$\hat{f}(\xi)=(\mathcal{F}f)(\xi)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbb{R}^{3}}f(x)e^{-i\xi\cdot
x}\mathrm{d}x~{}(\xi\in\mathbb{R}^{3}).$

The Fourier inverse transform is denoted by $\mathcal{F}^{-1}$:

$(\mathcal{F}^{-1}f)(x)=\frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbb{R}^{3}}f(\xi)e^{i\xi\cdot
x}\mathrm{d}\xi~{}(x\in\mathbb{R}^{3}).$

We recall the Sobolev inequalities.

######  Lemma 2.1.

The following inequalities hold:

$\displaystyle\mathrm{(i)}~{}\|u\|_{L^{p}}$ $\displaystyle\leq
C\|u\|_{H^{1}}~{}\mathrm{for}~{}2\leq p\leq 6,~{}u\in H^{1}.$
$\displaystyle\mathrm{(ii)}~{}\|u\|_{L^{p}}$ $\displaystyle\leq
C\|u\|_{H^{2}}~{}\mathrm{for}~{}2\leq p\leq\infty,~{}u\in H^{2}.$

We next introduce the following elementary inequality to control the Duammel
term.

######  Lemma 2.2.

The following estimates hold:

$\mathrm{(i)}~{}\int_{0}^{t}(1+t-s)^{\frac{1}{2}}(1+s)^{-2}\mathrm{d}s\leq
C(1+t)^{\frac{1}{2}},~{}t\geq 0,$

Here $C$ is a positive constant independent of $t$.

##  3 Main Result

In this section, we state the main result of this paper.

We set $u(t)=(\phi(t),w(t),G(t))=(\rho(t)-1,v(t),F(t)-I).$ Then $u(t)$
satisfies the following initial value problem

$\left\\{\begin{array}[]{l}\partial_{t}\phi+\mathrm{div}w=g_{1}(u),\\\\[4.30554pt]
\partial_{t}w-\nu{\Delta}w-\tilde{\nu}\nabla\mathrm{div}w+\gamma^{2}\nabla\phi-\beta^{2}\mathrm{div}G=g_{2}(u),\\\\[4.30554pt]
\partial_{t}G-\nabla w=g_{3}(u),\\\\[4.30554pt]
\nabla\phi+\mathrm{div}{}^{\top}\\!G=g_{4}(u),\\\\[4.30554pt]
u|_{t=0}=u_{0}=(\phi_{0},w_{0},G_{0}).\end{array}\right.$ (3.1)

Here $g_{j}(u),j=1,2,3,4,$ denote the nonlinear terms;

$\displaystyle\begin{array}[]{l}g_{1}(u)=-\mathrm{div}(\phi w),\\\\[4.30554pt]
\displaystyle g_{2}(u)=-w\cdot\nabla w+\frac{\phi}{1+\phi}(-\nu\Delta
w-\tilde{\nu}\nabla\mathrm{div}w+\gamma^{2}\nabla\phi)-\frac{1}{1+\phi}\nabla
R(\phi)\\\\[8.61108pt]
\displaystyle\quad\quad-\frac{\beta^{2}\phi}{1+\phi}\mathrm{div}G+\frac{\beta^{2}}{1+\phi}\mathrm{div}(\phi
G+G{}^{\top}\\!G+\phi G{}^{\top}\\!G),\\\\[8.61108pt] g_{3}(u)=-w\cdot\nabla
G+\nabla wG,\\\\[4.30554pt]
g_{4}(u)=g_{4}(\phi,G)=-\mathrm{div}(\phi{}^{\top}\\!G),\\\\[4.30554pt]
\end{array}$

where

$R(\phi)=\phi^{2}\int_{0}^{1}{P}^{\prime\prime}(1+s\phi)\mathrm{d}s,~{}\nabla
R=O(\phi)\nabla\phi$

for $|\phi|\ll 1$.

The following proposition ensures the global in time existence and the $L^{2}$
decay estimates of solutions.

######  Proposition 3.1.

$([6,12])$ Let $u_{0}\in H^{N},~{}N\geq 3$. There is a positive number
$\epsilon_{0}$ such that if $u_{0}$ satisfies
$\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{3}}\leq\epsilon_{0}$, then there exists a
unique solution $u(t)\in C([0,\infty);H^{N})$ of the problem $(3.1)$, and
$u(t)=(\phi(t),w(t),G(t))$ satisfies

$\displaystyle\|u(t)\|_{H^{N}}^{2}+\int_{0}^{t}(\|\nabla\phi(s)\|_{H^{N-1}}^{2}+\|\nabla
w(s)\|_{H^{N}}^{2}+\|\nabla G(s)\|_{H^{N-1}}^{2})\mathrm{d}s\leq
C_{N}\|u_{0}\|_{H^{N}}^{2},$ $\displaystyle\|\nabla^{k}u(t)\|_{L^{2}}\leq
C(1+t)^{-\frac{3}{4}-\frac{k}{2}}(\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{N}})$

for $k=0,1,2,\ldots,N-1$ and $t\geq 0$.

In addition, if there exists a positive number $r>0$ such that the following
condition satisfies

$\displaystyle|\hat{\phi}_{0}(\xi)|>c_{0},|\hat{m}_{0}(\xi)|+|\hat{\mathcal{G}}_{0}(\xi)-{}^{\top}\hat{\mathcal{G}}_{0}(\xi)|\leq
c_{1}|\xi|^{\eta_{0}}$ (3.2)

for $0\leq|\xi|\leq r$, where
$(m_{0},\mathcal{G}_{0}):=(\rho_{0}v_{0},\rho_{0}F_{0}-I)$; $c_{0}$, $c_{1}$
and $\eta_{0}$ are positive numbers independent of $t$, the following lower
$L^{2}$ estimate holds

$\displaystyle\|u(t)\|_{L^{2}}\geq c(1+t)^{-\frac{3}{4}}$ (3.3)

uniformly for $t\geq R_{1}$. Here $R_{1}$ is a large positive number, and $c$
is a positive number independent of $t$.

We next state the main result of this paper which gives the upper estimate of
the $L^{1}$ norm of solutions.

######  Theorem 3.2.

Assume that $\phi_{0}$ and $G_{0}$ satisfy
$\nabla\phi_{0}+\mathrm{div}{}^{\top}\\!G_{0}=g_{4}(\phi_{0},G_{0})$ and
$(I+G_{0})^{-1}=\nabla X_{0}$ for some vector field $X_{0}$. There is a
positive number $\epsilon$ such that if $u_{0}=(\phi_{0},w_{0},G_{0})$
satisfies $\|u_{0}\|_{H^{4}}+\|u_{0}\|_{L^{1}}\leq\epsilon$ and $u_{0}\in
W^{2,1}$, then there exists a unique solution $u(t)\in C([0,\infty);H^{4})$ of
the problem $(3.1)$ satisfying

$\displaystyle\|u(t)\|_{L^{1}}\leq
C(1+t)^{\frac{1}{2}}(\|u_{0}\|_{W^{2,1}}+\|u_{0}\|_{H^{4}})$

uniformly for $t>0$. Here $C$ is a positive constant.

We also prove the lower $L^{1}$ estimate to show that the obtained rate
$(1+t)^{\frac{1}{2}}$ is sharp.

######  Theorem 3.3.

Under the assumptions in Proposition 3.1 and Theorem 3.2, there exists a
positive large number $R_{1}$ such that the lower $L^{1}$ estimate of the
solution $u$ of the problem $(3.1)$ holds

$\displaystyle\|u(t)\|_{L^{1}}\geq c(1+t)^{\frac{1}{2}}$

uniformly for $t\geq R_{1}$. Here $c$ is a positive constant independent of
time $t$.

Proof of Theorem 3.3. Since we assume the condition $(3.2)$, the lower $L^{2}$
estimate $(3.3)$ holds uniformly for $t\geq R_{1}$. Here $R_{1}$ is taken in
Proposition 3.1. We next introduce the following $L^{\infty}$ decay estimate
which can be proved in a similar manner to [9, Theorem 3.2 (i)] and Theorem
3.2

$\displaystyle\|u(t)\|_{L^{\infty}}\leq
C(1+t)^{-2}(\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{3}}),~{}t\geq 0.$ (3.4)

By using $(3.3)$, $(3.4)$ and the interpolation inequality
$\|u(t)\|_{L^{2}}\leq\|u(t)\|_{L^{1}}^{\frac{1}{2}}\|u(t)\|_{L^{\infty}}^{\frac{1}{2}}$,
we have

$\displaystyle c(1+t)^{-\frac{3}{4}}$ $\displaystyle\leq\|u(t)\|_{L^{2}}$
$\displaystyle\leq\|u(t)\|_{L^{\infty}}^{\frac{1}{2}}\|u(t)\|_{L^{1}}^{\frac{1}{2}}$
$\displaystyle\leq C(1+t)^{-1}\|u(t)\|_{L^{1}}^{\frac{1}{2}}.$

This yields

$\|u(t)\|_{L^{1}}\geq c(1+t)^{\frac{1}{2}}.$

This complete the proof. $\blacksquare$

##  4 Proof of Theorem 3.2

This section is devoted to the proof of Theorem 3.2.

The global existence of solutions is proved by Proposition 3.1. Hence we focus
on the derivation of the $L^{1}$ estimate.

As we mentioned above, it is not valid to apply the semigroup theory to the
problem $(3.1)$ directly. To overcome this difficulty, we reformulate the
problem $(3.1)$.

Let $x=x(X,t)$ be a material coordinate which solves the following flow map

$\left\\{\begin{array}[]{l}\displaystyle\frac{dx}{dt}(X,t)=v(x(X,t),t),\\\\[4.30554pt]
x(X,0)=X.\end{array}\right.$

We next define $X=X(x,t)$ as the inverse of $x=x(X,t)$ and $\psi=x-X$.
According to [1,21], $F$ is defined as $F=\frac{\partial x}{\partial X}$. It
is shown in [19] that its inverse $F^{-1}$ is written as $F^{-1}(x,t)=\nabla
X(x,t)$ if $F_{0}^{-1}$ has the form $F^{-1}_{0}=\nabla X_{0}$ with some
vector field $X_{0}$. Then $\psi$ solves

$\displaystyle\partial_{t}\psi-v=-v\cdot\nabla\psi,$ (4.1)

and satisfies

$G=\nabla\psi+h(\nabla\psi),$ (4.2)

where $h(\nabla\psi)=(I-\nabla\psi)^{-1}-I-\nabla\psi$.

We note that $(4.2)$ is equivalent to

$\nabla\psi=I-(I+G)^{-1}.$ (4.3)

We next read the constraint $(1.8)$ as

$1+\phi=\rho=\mathrm{det}F^{-1}=\mathrm{det}(I+G)^{-1}.$

By using $(4.3)$ and the following expansion for a $3\times 3$ matrix $A$

$\mathrm{det}(I+A)=1+\mathrm{tr}A+\frac{1}{2}(\mathrm{tr}A^{2}-(\mathrm{tr}A)^{2})+\mathrm{det}A,$

we have

$\displaystyle\phi=-\mathrm{div}\psi+\frac{1}{2}(\mathrm{tr}(\nabla\psi)^{2}-(\mathrm{tr}(\nabla\psi))^{2})-\mathrm{det}(\nabla\psi).$
(4.4)

We set $\psi_{0}=\psi|_{t=0}$. The following estimates hold for $\phi$, $G$
and $\nabla\psi$.

######  Lemma 4.1.

Assume that $G$ and $\psi$ satisfy $(4.2)$. There is a positive number
$\epsilon_{1}<1$ such that if $\|G\|_{C([0,\infty);H^{3})}\leq\epsilon_{1}$,
the following inequalities hold uniformly for $t\geq 0$:

$\displaystyle C^{-1}\|\nabla\psi(t)\|_{L^{p}}\leq\|G(t)\|_{L^{p}}\leq
C\|\nabla\psi(t)\|_{L^{p}},~{}p=1,2,$ (4.5)
$\displaystyle\|\nabla^{2}\psi(t)\|_{L^{2}}\leq C\|\nabla G(t)\|_{L^{2}},$
(4.6) $\displaystyle\|\nabla^{3}\psi(t)\|_{L^{2}}\leq C(\|\nabla
G(t)\|_{H^{1}}^{2}+\|\nabla^{2}G(t)\|_{L^{2}}),$ (4.7)
$\displaystyle\|\nabla^{4}\psi(t)\|_{L^{2}}\leq C(\|\nabla
G(t)\|_{H^{1}}\|\nabla^{2}G(t)\|_{H^{1}}+\|\nabla^{3}G(t)\|_{L^{2}}).$ (4.8)
$\displaystyle\|\nabla^{2}\psi_{0}\|_{L^{1}}\leq C\|\nabla G_{0}\|_{L^{1}},$
(4.9) $\displaystyle\|\nabla^{3}\psi_{0}\|_{L^{1}}\leq
C\|\nabla^{2}G_{0}\|_{L^{1}},$ (4.10)
$\displaystyle\|\phi(t)\|_{L^{1}}\leq\|\mathrm{div}\psi(t)\|_{L^{1}}+C(1+\|\nabla\psi(t)\|_{L^{\infty}})\|\nabla\psi(t)\|_{L^{2}}^{2}.$
(4.11)

Proof. The inequalities $(4.5)$–$(4.8)$ are shown in [9, Lemma 4.1]. The
inequalities $(4.9)$ and $(4.10)$ are established in a similar argument as in
the proof of (4.6). We only derive $(4.11)$ here.

In order to obtain $(4.11)$, we use $(4.4)$. Since

$\|(\mathrm{tr}(\nabla\psi)^{2}-(\mathrm{tr}(\nabla\psi))^{2})\|_{L^{1}}\leq
C\|\nabla\psi\|_{L^{2}}^{2},$ $\|\mathrm{det}(\nabla\psi)\|_{L^{1}}\leq
C\|\nabla\psi\|_{L^{\infty}}\|\nabla\psi\|_{L^{2}}^{2},$

we have

$\|\phi\|_{L^{1}}\leq\|\mathrm{div}\psi\|_{L^{1}}+C(1+\|\nabla\psi\|_{L^{\infty}})\|\nabla\psi\|_{L^{2}}^{2}.$

This completes the proof of Lemma 4.1. $\blacksquare$

Let $U$ and $U_{0}$ be
$U=(\tilde{\phi},w,\tilde{G})=(-\mathrm{div}\psi,w,\nabla\psi)$ and
$U_{0}=(\tilde{\phi}_{0},w_{0},\tilde{G}_{0})=(-\mathrm{div}\psi_{0},w_{0},\nabla\psi_{0})$,
respectively. By taking $\epsilon_{0}$ in Proposition 3.1 such that
$C_{3}\epsilon_{0}\leq\epsilon_{1}$, thanks to Lemma 4.1, the behavior of $u$
is identified from $U$: $\|u(t)\|_{L^{1}}\leq C\|U(t)\|_{L^{1}}$. By coupling
$(3.1)$ and $(4.1)$, we arrive at the following problem for $U$

$\left\\{\begin{array}[]{l}\partial_{t}\tilde{\phi}+\mathrm{div}w=N_{1}(U),\\\
\partial_{t}w-\nu{\Delta}w-\tilde{\nu}\nabla\mathrm{div}w+\gamma^{2}\nabla\tilde{\phi}-\beta^{2}\mathrm{div}\tilde{G}=N_{2}(U),\\\
\partial_{t}\tilde{G}-\nabla w=N_{3}(U),\\\
\tilde{\phi}+\mathrm{tr}\tilde{G}=0,~{}\tilde{G}=\nabla\psi,\\\
U|_{t=0}=U_{0}.\end{array}\right.$ (4.12)

Here $N_{j}(U),j=1,2,3,$ denote nonlinear terms;

$\displaystyle\begin{array}[]{l}N_{1}(U)=\mathrm{div}(w\cdot\nabla\psi),\\\\[4.30554pt]
\displaystyle
N_{2}(U)=g_{2}(u)-\frac{\gamma^{2}}{2}\nabla(\mathrm{tr}(\nabla\psi)^{2}-(\mathrm{tr}(\nabla\psi))^{2})-\gamma^{2}\nabla\mathrm{det}(\nabla\psi)+\beta^{2}\mathrm{div}h(\nabla\psi),\\\\[8.61108pt]
N_{3}(U)=-\nabla(w\cdot\nabla\psi).\end{array}$

We note that $N_{1}(U)$ and $N_{3}(U)$ also satisfy the same linear constraint
as $\tilde{\phi}$ and $\tilde{G}$

$\displaystyle N_{1}(U)+\mathrm{tr}N_{3}(U)=0.$ (4.13)

In what follows, we omit tildes $\tilde{\cdot}$ of $\tilde{\phi}$ and
$\tilde{G}$ for simplicity. The problem $(4.12)$ is reduced to

$\left\\{\begin{array}[]{l}\partial_{t}U+LU=N(U),\\\
\phi+\mathrm{tr}G=0,~{}G=\nabla\psi,\\\ U|_{t=0}=U_{0},\end{array}\right.$
(4.14)

where

$L=\left(\begin{array}[]{@{\ }cc@{\ }cc@{\ } }0&\mathrm{div}&0\\\
\gamma^{2}\nabla&-\nu\Delta-\tilde{\nu}\nabla\mathrm{div}&-\beta^{2}\mathrm{div}\\\
0&-\nabla&0\end{array}\right),~{}N(U)=\left(\begin{array}[]{l}N_{1}(U)\\\
N_{2}(U)\\\ N_{3}(U)\end{array}\right).$

By using the Duammel principle, $U$ satisfies the following integral equations

$\displaystyle U(t)=e^{-tL}U_{0}+\int_{0}^{t}e^{-(t-s)L}N(U(s))\mathrm{d}s.$
(4.15)

To analyize the linearized semigroup $e^{-tL}$, we introduce several
notations. We set

$\displaystyle Q$
$\displaystyle=I-P=\mathcal{F}^{-1}\frac{\xi{}^{\top}\xi}{|\xi|^{2}}\mathcal{F},$
$\displaystyle\mathcal{K}_{t}^{\lambda}(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\right](x),$
$\displaystyle\mathcal{K}_{t}^{\mu}(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\right](x).$

Here $\lambda_{\pm}(\xi)$ and $\mu_{\pm}(\xi)$ are given by

$\displaystyle\lambda_{\pm}(\xi)$
$\displaystyle=\frac{-\nu|\xi|^{2}\pm\sqrt{\nu^{2}|\xi|^{4}-4\beta^{2}|\xi|^{2}}}{2},$
$\displaystyle\mu_{\pm}(\xi)$
$\displaystyle=\frac{-(\nu+\tilde{\nu})|\xi|^{2}\pm\sqrt{(\nu+\tilde{\nu})^{2}|\xi|^{4}-4(\beta^{2}+\gamma^{2})|\xi|^{2}}}{2}.$

We see that the following properties of $\lambda_{\pm}(\xi)$ and
$\mu_{\pm}(\xi)$ hold:

$\displaystyle\lambda_{\pm}(\xi)\sim-\frac{\nu}{2}|\xi|^{2}\pm
i\beta|\xi|,~{}\mathrm{for}~{}|\xi|\ll 1,$
$\displaystyle\lambda_{+}(\xi)\sim-\frac{\beta^{2}}{\nu},~{}\lambda_{-}(\xi)\sim-\nu|\xi|^{2},~{}\mathrm{for}~{}|\xi|\gg
1,$ $\displaystyle\mu_{\pm}(\xi)\sim-\frac{\nu+\tilde{\nu}}{2}|\xi|^{2}\pm
i\sqrt{\beta^{2}+\gamma^{2}}|\xi|,~{}\mathrm{for}~{}|\xi|\ll 1,$
$\displaystyle\mu_{+}(\xi)\sim-\frac{\beta^{2}+\gamma^{2}}{\nu+\tilde{\nu}},~{}\mu_{-}(\xi)\sim-(\nu+\tilde{\nu})|\xi|^{2},~{}\mathrm{for}~{}|\xi|\gg
1.$

According to [9], the expression of the semigroup
$U(t)={}^{\top}(\phi(t),w(t),G(t))=e^{-tL}U_{0}$ is given by

$\displaystyle\left(\begin{array}[]{l}\phi(x,t)\\\ w(x,t)\\\
G(x,t)\end{array}\right)=\left(\begin{array}[]{@{\ }cc@{\ }cc@{\ }
}\mathcal{K}^{11}(t)&\mathcal{K}^{12}(t)&\mathcal{K}^{13}(t)\\\
\mathcal{K}^{21}(t)&\mathcal{K}^{22}(t)&\mathcal{K}^{23}(t)\\\
\mathcal{K}^{31}(t)&\mathcal{K}^{32}(t)&\mathcal{K}^{33}(t)\end{array}\right)\left(\begin{array}[]{l}\phi_{0}(x)\\\
w_{0}(x)\\\ G_{0}(x)\end{array}\right),$ (4.25)

provided that $\phi_{0}+\mathrm{tr}G_{0}=0$ and $G_{0}=\nabla\psi_{0}$.

Here $\mathcal{K}^{jk}(t),~{}j,k=1,2,3,$ are the linear operators defined as

$\displaystyle\mathcal{K}^{11}(t)\phi_{0}(x)$
$\displaystyle=(\partial_{t}-(\nu+\tilde{\nu})\Delta)(\mathcal{K}_{t}^{\mu}*\phi_{0})(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{\mu_{+}(\xi)e^{\mu_{-}(\xi)t}-\mu_{-}(\xi)e^{\mu_{+}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\hat{\phi}_{0}(\xi)\right](x),$
$\displaystyle\mathcal{K}^{12}(t)w_{0}(x)$
$\displaystyle=-\mathrm{div}(\mathcal{K}_{t}^{\mu}*w_{0})(x)$
$\displaystyle=i\mathcal{F}^{-1}\left[\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\xi\cdot\hat{w}_{0}(\xi)\right](x),$
$\displaystyle\mathcal{K}^{13}(t)G_{0}(x)$ $\displaystyle=0,$
$\displaystyle\mathcal{K}^{21}(t)\phi_{0}(x)$
$\displaystyle=-\gamma^{2}\nabla(\mathcal{K}_{t}^{\mu}*\phi_{0})(x),$
$\displaystyle=i\gamma^{2}\mathcal{F}^{-1}\left[\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\hat{\phi}_{0}(\xi)\xi\right](x),$
$\displaystyle\mathcal{K}^{22}(t)w_{0}(x)$
$\displaystyle=\partial_{t}(\mathcal{K}_{t}^{\lambda}*Pw_{0})(x)+\partial_{t}(\mathcal{K}_{t}^{\mu}*Qw_{0})(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{\lambda_{+}(\xi)e^{\lambda_{+}(\xi)t}-\lambda_{-}(\xi)e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\left(I-\frac{\xi^{\top}\xi}{|\xi|^{2}}\right)\hat{w}_{0}(\xi)\right](x)$
$\displaystyle\quad+\mathcal{F}^{-1}\left[\frac{\mu_{+}(\xi)e^{\mu_{+}(\xi)t}-\mu_{-}(\xi)e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\frac{\xi^{\top}\xi}{|\xi|^{2}}\hat{w}_{0}(\xi)\right](x),$
$\displaystyle\mathcal{K}^{23}(t)G_{0}(x)$
$\displaystyle=\beta^{2}(PG_{0}*\nabla\mathcal{K}_{t}^{\lambda})(x)+\beta^{2}(QG_{0}*\nabla\mathcal{K}_{t}^{\mu})(x)$
$\displaystyle=-i\beta^{2}\mathcal{F}^{-1}\left[\frac{e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\left(I-\frac{\xi^{\top}\xi}{|\xi|^{2}}\right)\hat{G}_{0}(\xi)\xi\right](x)$
$\displaystyle\quad\quad-i\beta^{2}\mathcal{F}^{-1}\left[\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\frac{\xi^{\top}\xi}{|\xi|^{2}}\hat{G}_{0}(\xi)\xi\right](x),$
$\displaystyle\mathcal{K}^{31}(t)\phi_{0}(x)$ $\displaystyle=0,$
$\displaystyle\mathcal{K}^{32}(t)w_{0}(x)$
$\displaystyle=(Pw_{0}*\nabla\mathcal{K}_{t}^{\lambda})(x)+(Qw_{0}*\nabla\mathcal{K}_{t}^{\mu})(x)$
$\displaystyle=-i\mathcal{F}^{-1}\left[\frac{e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\left(I-\frac{\xi^{\top}\xi}{|\xi|^{2}}\right)\hat{w}_{0}(\xi){}^{\top}\xi\right](x)$
$\displaystyle\quad\quad-i\mathcal{F}^{-1}\left[\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\frac{\xi^{\top}\xi}{|\xi|^{2}}\hat{w}_{0}(\xi){}^{\top}\xi\right](x),$
$\displaystyle\mathcal{K}^{33}(t)G_{0}(x)$
$\displaystyle=(\partial_{t}-\nu\Delta)(\mathcal{K}_{t}^{\lambda}*PG_{0})(x)$
$\displaystyle\quad+(\partial_{t}-(\nu+\tilde{\nu})\Delta)(\mathcal{K}_{t}^{\mu}*QG_{0})(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{\lambda_{+}(\xi)e^{\lambda_{-}(\xi)t}-\lambda_{-}(\xi)e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\left(I-\frac{\xi^{\top}\xi}{|\xi|^{2}}\right)\hat{G}_{0}(\xi)\right](x)$
$\displaystyle\quad\quad+\mathcal{F}^{-1}\left[\frac{\mu_{+}(\xi)e^{\mu_{-}(\xi)t}-\mu_{-}(\xi)e^{\mu_{+}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\frac{\xi^{\top}\xi}{|\xi|^{2}}\hat{G}_{0}(\xi)\right](x).$

In view of the asymptotic profiles of $\lambda_{\pm}(\xi)$ and
$\mu_{\pm}(\xi)$ in the Fourier space, we decompose the solution $U(t)$ of the
problem $(4.14)$ into its low and high frequency parts. Let
$\hat{\varphi}_{1},~{}\hat{\varphi}_{M},~{}\hat{\varphi}_{\infty}\in
C^{\infty}(\mathbb{R}^{3};[0,1])$ be cut-off functions such that

$\displaystyle\hat{\varphi}_{1}(\xi)$
$\displaystyle=\begin{cases}1&|\xi|\leq\frac{M_{1}}{2},\\\
0&|\xi|\geq\frac{M_{1}}{\sqrt{2}},\end{cases}~{}\hat{\varphi}_{1}(-\xi)=\hat{\varphi}_{1}(\xi),$
$\displaystyle\hat{\varphi}_{\infty}(\xi)$
$\displaystyle=\begin{cases}1&|\xi|\geq 2M_{2}\\\
0&|\xi|\leq\sqrt{2}M_{2},\end{cases}~{}\hat{\varphi}_{\infty}(-\xi)=\hat{\varphi}_{\infty}(\xi),$
$\displaystyle\hat{\varphi}_{M}(\xi)$
$\displaystyle=1-\hat{\varphi}_{1}(\xi)-\hat{\varphi}_{\infty}(\xi),$

where

$M_{1}=\min\left\\{\frac{\beta}{\nu},\frac{\sqrt{\beta^{2}+\gamma^{2}}}{\nu+\tilde{\nu}}\right\\},~{}M_{2}=\max\left\\{\frac{\beta}{\nu},\frac{\sqrt{\beta^{2}+\gamma^{2}}}{\nu+\tilde{\nu}}\right\\}.$

We define operators $P_{j},~{}j=1,\infty,$ on $L^{2}$ as

$P_{1}u=\mathcal{F}^{-1}(\hat{\varphi}_{1}\hat{u}),~{}P_{\infty}u=\mathcal{F}^{-1}((\hat{\varphi}_{M}+\hat{\varphi}_{\infty})\hat{u})~{}\mathrm{for}~{}u\in
L^{2}.$

######  Lemma 4.2.

$P_{j}~{}(j=1,\infty)$ have the following properties.

$(i)$ $P_{1}+P_{\infty}=I$.

$(ii)$ $\partial_{x}^{\alpha}P_{1}=P_{1}\partial_{x}^{\alpha}$,
$\|\partial_{x}^{\alpha}P_{1}f\|_{L^{2}}\leq C_{\alpha}\|f\|_{L^{2}}$ for
$\alpha\in(\\{0\\}\cup\mathbb{N})^{3}$ and $f\in L^{2}$.

$(iii)$ $\partial_{x}^{\alpha}P_{\infty}=P_{\infty}\partial_{x}^{\alpha}$,
$\|\partial_{x}^{\alpha}P_{\infty}f\|_{L^{2}}\leq
C\|\nabla\partial_{x}^{\alpha}P_{\infty}f\|_{L^{2}}$ for
$\alpha\in(\\{0\\}\cup\mathbb{N})^{3}$ with $|\alpha|=k\geq 0$ and $f\in
H^{k+1}$.

Lemma 4.2 immediately follows from the definitions of $P_{j},~{}j=1,\infty,$
and the Plancherel theorem. We omit the proof.

The solution $U(t)$ of $(4.14)$ is decomposed as

$U(t)=U_{1}(t)+U_{\infty}(t),~{}U_{1}(t)=P_{1}U(t),~{}U_{\infty}(t)=P_{\infty}U(t).$

By applying $P_{j}$ to $(4.15)$,
$U_{j}(t)=(\phi_{j}(t),w_{j}(t),G_{j}(t)),~{}j=1,\infty,$ satisfy

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
U_{j}(t)=e^{-tL}U_{j}(0)+\int_{0}^{t}e^{-(t-s)L}P_{j}N(U(s))\mathrm{d}s,\\\\[6.45831pt]
\phi_{j}+\mathrm{tr}G_{j}=0,~{}P_{j}N_{1}(U)+\mathrm{tr}P_{j}N_{3}=0,\\\\[6.45831pt]
U_{j}|_{t=0}=P_{j}U_{0}.\end{array}\right.$ (4.29)

Concerning to the $L^{1}$ estimate of $e^{-tL}U_{0}$, we have the following
proposition.

######  Proposition 4.3.

Let $\phi_{0}+\mathrm{tr}G_{0}=0$ and $G_{0}=\nabla\psi_{0}$. Then the
following estimates hold for $t>0$:

$\displaystyle(i)\hskip 12.91663pt\|e^{-tL}P_{1}U_{0}\|_{L^{1}}$
$\displaystyle\leq C(1+t)^{\frac{1}{2}}\|U_{0}\|_{L^{1}},$
$\displaystyle(ii)~{}\|e^{-tL}P_{\infty}U_{0}\|_{L^{1}}$ $\displaystyle\leq
Ce^{-ct}\|U_{0}\|_{W^{2,1}}.$

The estimate $(i)$ is done in [9, Lemma 6.4]. For the estimate $(ii)$, we will
discuss in Section 5.

In order to estimate the Duammel terms
$\int_{0}^{t}e^{-(t-s)L}P_{j}N(U(s))\mathrm{d}s,~{}j=1,\infty$, we give the
following $L^{2}$ decay estimates for $\nabla^{k}U(t)$.

######  Proposition 4.4.

There exists a positive number $\epsilon_{1}$ such that if
$\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{4}}\leq\epsilon_{1}$, then the following
inequality holds for $k=0,1,2,3$ and $t\geq 0$:

$\|\nabla^{k}U(t)\|_{L^{2}}\leq
C(1+t)^{-\frac{3}{4}-\frac{k}{2}}(\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{4}}),$

Proposition 4.4 follows from Proposition 3.1 and Lemma 4.1.

The estimates of
$\int_{0}^{t}\|e^{-(t-s)L}P_{j}N(U(s))\|_{L^{1}}\mathrm{d}s,j=1,\infty,$ are
given as follows.

######  Lemma 4.5.

There exists a positive number $\epsilon_{1}$ such that if $u_{0}\in L^{1}\cap
H^{4}$ and $\|u_{0}\|_{H^{4}}\leq\epsilon_{1}$, then the following estimates
hold:

$\int_{0}^{t}\|e^{-(t-s)L}P_{j}N(U(s))\|_{L^{1}}\mathrm{d}s\leq
C(1+t)^{\frac{1}{2}}(\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{4}}),~{}j=1,\infty,~{}t\geq
0.$

Proof. We first show the case $j=1$. We obtain the following estimate in a
similar argument as in the proof of Lemma 4.3:

$\displaystyle\begin{array}[]{l}\displaystyle\left\|e^{-(t-s)L}P_{1}N(U(s))\right\|_{L^{1}}\leq
C(1+t-s)^{\frac{1}{2}}\|N(U(s))\|_{L^{1}}.\end{array}$ (4.31)

In view of Lemma 2.1 and Proposition 4.4, we have

$\displaystyle\begin{array}[]{l}\|N(U(s))\|_{L^{1}}\leq
C\|U(s)\|_{H^{2}}\|\nabla U(s)\|_{H^{1}}\leq
C(1+s)^{-2}(\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{4}}).\end{array}$ (4.33)

We see from Lemma 2.2, $(4.18)$ and $(4.19)$ that

$\displaystyle\begin{array}[]{l}\displaystyle\int_{0}^{t}\|e^{-(t-s)L}P_{1}N(U(s))\|_{L^{1}}\mathrm{d}s\\\\[8.61108pt]
\quad\displaystyle\leq\displaystyle
C\int_{0}^{t}(1+t-s)^{\frac{1}{2}}\|N(U(s))\|_{L^{1}}\mathrm{d}s\\\\[8.61108pt]
\quad\displaystyle\leq
C\int_{0}^{t}(1+t-s)^{\frac{1}{2}}(1+s)^{-2}\mathrm{d}s(\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{4}})\\\\[8.61108pt]
\quad\displaystyle\leq
C(1+t)^{\frac{1}{2}}(\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{4}}).\end{array}$ (4.38)

We next consider the case $j=\infty$.

We obtain the following estimate in a similar argument as in the proof of
Lemma 4.3:

$\displaystyle\|e^{-(t-s)L}P_{\infty}N(U(s))\|_{L^{1}}\leq
Ce^{-c(t-s)}\|N(U(s))\|_{W^{1,1}}.$ (4.39)

In view of Lemma 2.1 and Proposition 4.4, we have

$\displaystyle\begin{array}[]{l}\|N(U(s))\|_{W^{2,1}}\leq
C(\|U(s)\|_{H^{3}}^{2}+\|u(s)\|_{L^{2}}\|\nabla^{4}w(s)\|_{L^{2}})\leq
C\|u_{0}\|_{H^{4}}.\end{array}$ (4.41)

It follows from $(4.21)$ and $(4.22)$ that

$\displaystyle\begin{array}[]{l}\displaystyle\int_{0}^{t}\|e^{-(t-s)L}P_{\infty}N(U(s))\|_{L^{1}}\mathrm{d}s\\\\[8.61108pt]
\quad\displaystyle\leq
C\int_{0}^{t}e^{-c(t-s)}\|N(U(s))\|_{W^{2,1}}\mathrm{d}s\\\\[8.61108pt]
\quad\displaystyle\leq C\|u_{0}\|_{H^{4}}\\\\[8.61108pt]
\quad\displaystyle\leq C(1+t)^{\frac{1}{2}}\|u_{0}\|_{H^{4}}.\end{array}$

This completes the proof. $\blacksquare$

Proof of Theorem 3.2. By taking $L^{1}$ norm of the first equation of
$(4.17)$, we have

$\displaystyle\|U_{j}(t)\|_{L^{1}}$
$\displaystyle\leq\|e^{-tL}U_{j}(0)\|_{L^{1}}+\int_{0}^{t}\|e^{-(t-s)L}P_{j}N(s)\|_{L^{1}}\mathrm{d}s.$
(4.42)

Combining Lemma 4.3, Lemma 4.5 and $(4.23)$, we arrive at

$\|U(t)\|_{L^{1}}\leq
C(1+t)^{\frac{1}{2}}(\|u_{0}\|_{W^{2,1}}+\|u_{0}\|_{H^{4}}),~{}t\geq 0.$

By using Lemma 4.1, we obtain

$\|u(t)\|_{L^{1}}\leq C\|U(t)\|_{L^{1}}\leq
C(1+t)^{\frac{1}{2}}(\|u_{0}\|_{W^{2,1}}+\|u_{0}\|_{H^{4}}),~{}t\geq 0.$

This completes the proof of Theorem 3.2. $\blacksquare$

##  5 Proof of Proposition 4.3 (ii).

In this section, we prove Proposition 4.3 (ii).

For $j=M,\infty$, we set

$\displaystyle\mathcal{K}_{j}^{\lambda_{\pm}}(t)f(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{e^{\lambda_{\pm}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{j}(\xi)\hat{f}(\xi)\right],$
$\displaystyle\mathcal{M}_{j}^{\lambda_{\pm}}(t)f(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{\lambda_{\mp}(\xi)e^{\lambda_{\pm}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{j}(\xi)\hat{f}(\xi)\right],$
$\displaystyle\mathcal{K}_{j}^{\mu_{\pm}}(t)f(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{e^{\mu_{\pm}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\hat{\varphi}_{j}(\xi)\hat{f}(\xi)\right],$
$\displaystyle\mathcal{M}_{j}^{\mu_{\pm}}(t)f(x)$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{\mu_{\mp}(\xi)e^{\mu_{\pm}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\hat{\varphi}_{j}(\xi)\hat{f}(\xi)\right].$

We first consider the high frequency part.

######  Lemma 5.1.

The following estimates hold for $\alpha\in(\mathbb{N}\cup\\{0\\})^{3}$,
$j\geq k\geq 0$ and $t>0$:

$\displaystyle(i)~{}\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{+}}(t)f]\right\|_{L^{1}}+\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\mu_{+}}(t)f]\right\|_{L^{1}}\leq
Ce^{-ct}\|f\|_{W^{(|\alpha|-1)^{+},1}},$ (5.1)
$\displaystyle(ii)~{}\begin{array}[]{l}\displaystyle\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{-}}(t)f]\right\|_{L^{1}}+\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\mu_{-}}(t)f]\right\|_{L^{1}}\\\\[4.30554pt]
\displaystyle\quad\leq
Ce^{-ct}t^{-(j-k)}\|f\|_{W^{2k+(|\alpha|-1)^{+},1}},\end{array}$ (5.4)
$\displaystyle(iii)~{}\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{M}_{\infty}^{\lambda_{+}}(t)f]\right\|_{L^{1}}+\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{M}_{\infty}^{\mu_{+}}(t)f]\right\|_{L^{1}}\leq
Ce^{-ct}\|f\|_{W^{|\alpha|,1}},$ (5.5)
$\displaystyle(iv)~{}\begin{array}[]{l}\displaystyle\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{M}_{\infty}^{\lambda_{-}}(t)f]\right\|_{L^{1}}+\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{M}_{\infty}^{\mu_{-}}(t)f]\right\|_{L^{1}}\\\\[4.30554pt]
\displaystyle\quad\leq
Ce^{-ct}t^{-(j-k)}\|f\|_{W^{2k+(|\alpha|-1)^{+},1}},\end{array}$ (5.8)
$\displaystyle(v)~{}\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{+}}(t)Qf]\right\|_{L^{1}}+\left\|\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\mu_{+}}(t)Qf]\right\|_{L^{1}}\leq
Ce^{-ct}\|f\|_{W^{|\alpha|,1}},$ (5.9)
$\displaystyle(vi)~{}\begin{array}[]{l}\displaystyle\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{-}}(t)Qf]\right\|_{L^{1}}+\left\|\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\mu_{-}}(t)Qf]\right\|_{L^{1}}\\\\[4.30554pt]
\displaystyle\quad\leq
Ct^{-(j-k)}e^{-ct}\|f\|_{W^{2k+|\alpha|,1}}.\end{array}$ (5.12)

Here $a^{+}$ denotes $a^{+}=\max\\{0,a\\}$ for $a\in\mathbb{R}$.

Proof. We see from [20, Theorem 4.2.] that $(5.1)$–$(5.4)$ are true. Therefore
it remains to show $(5.5)$ and $(5.6)$.

We first write
$\partial_{t}^{j}\partial_{x}^{\alpha}\mathcal{K}_{\infty}^{\lambda_{\pm}}(t)Qf$
as

$\displaystyle\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{+}}(t)Qf]=\mathcal{F}^{-1}\left[\frac{\lambda_{+}(\xi)^{j}e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]*\partial_{x}^{\alpha}f.$

We use the formula

$e^{i\xi\cdot
x}=\sum_{|\eta|=m}\frac{(-ix)^{\eta}}{|x|^{2m}}\partial_{\xi}^{\eta}(e^{i\xi\cdot
x}).$ (5.13)

By $m$-times integration by parts and $(5.7)$, we have

$\displaystyle\mathcal{F}^{-1}\left[\frac{\lambda_{+}(\xi)^{j}e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]$
$\displaystyle=\frac{1}{(2\pi)^{\frac{3}{2}}}\sum_{|\eta|=m}\frac{(ix)^{\eta}}{|x|^{2m}}\int_{|\xi|\geq\sqrt{2}M_{2}}e^{i\xi\cdot
x}\partial_{\xi}^{\eta}\left[\frac{\lambda_{+}(\xi)^{j}e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]\mathrm{d}\xi.$

Since

$\displaystyle\left|\partial_{\xi}^{\eta}\left(\frac{\lambda_{+}(\xi)^{j}e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right)\right|\leq
C_{\eta}(1+t)^{|\eta|}e^{-c_{1}t}|\xi|^{-|\eta|-2},$

we obtain

$\displaystyle\begin{array}[]{l}\displaystyle\left|\mathcal{F}^{-1}\left[\frac{\lambda_{+}(\xi)^{j}e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]\right|\\\\[8.61108pt]
\displaystyle\quad\leq
C_{m}|x|^{-m}e^{-c_{1}t}\int_{|\xi|\geq\sqrt{2}M_{2}}|\xi|^{-m-2}\mathrm{d}\xi\\\\[8.61108pt]
\displaystyle\quad\leq C_{m}|x|^{-m}e^{-c_{1}t},~{}m\geq 2.\end{array}$ (5.17)

Therefore, by using $(5.8)$, we have

$\displaystyle\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{+}}(t)Qf]\|_{L^{1}}$
$\displaystyle\leq\left\|\mathcal{F}^{-1}\left[\frac{\lambda_{+}(\xi)^{j}e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]\right\|_{L^{1}}\|\partial_{x}^{\alpha}f\|_{L^{1}}$
$\displaystyle\leq Ce^{-c_{1}t}\int_{|x|\leq
1}|x|^{-2}\left(\int_{|\xi|\geq\sqrt{2}M_{2}}|\xi|^{-4}\mathrm{d}\xi\right)\mathrm{d}x\|f\|_{W^{|\alpha|,1}}$
$\displaystyle\quad+Ce^{-c_{1}t}\int_{|x|\geq
1}|x|^{-4}\left(\int_{|\xi|\geq\sqrt{2}M_{2}}|\xi|^{-6}\mathrm{d}\xi\right)\mathrm{d}x\|f\|_{W^{|\alpha|,1}}$
$\displaystyle\leq Ce^{-ct}\|f\|_{W^{|\alpha|,1}}.$

Similarly, we obtain

$\displaystyle\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\mu_{+}}(t)Qf]\|_{L^{1}}\leq
Ce^{-ct}\|f\|_{W^{|\alpha|,1}}.$

This completes the proof of $(5.5)$.

We next show $(5.6)$. We write
$\partial_{t}^{j}\partial_{x}^{\alpha}\mathcal{K}_{\infty}^{\lambda_{-}}(t)Qf$
as

$\displaystyle\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{-}}(t)Qf]$
$\displaystyle=\mathcal{F}^{-1}\left[\frac{\lambda_{-}(\xi)^{j}e^{\lambda_{-}(\xi)t}(1+|\xi|^{2})^{j-k}}{(\lambda_{+}(\xi)-\lambda_{-}(\xi))(1+|\xi|^{2})^{j}}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]*\partial_{x}^{\alpha}(1-\Delta)^{k}f,$

where $(1-\Delta)^{k}=\mathcal{F}^{-1}(1+|\xi|^{2})^{k}\mathcal{F}$.

By $m$-times integration by parts and $(5.7)$, we have

$\displaystyle\mathcal{F}^{-1}\left[\frac{\lambda_{-}(\xi)^{j}e^{\lambda_{-}(\xi)t}(1+|\xi|^{2})^{j-k}}{(\lambda_{+}(\xi)-\lambda_{-}(\xi))(1+|\xi|^{2})^{j}}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]$
$\displaystyle=\frac{1}{(2\pi)^{\frac{3}{2}}}\sum_{|\eta|=m}\frac{(ix)^{\eta}}{|x|^{2m}}\int_{|\xi|\geq\sqrt{2}M_{2}}e^{i\xi\cdot
x}\partial_{\xi}^{\eta}\left[\frac{\lambda_{-}(\xi)^{j}e^{\lambda_{-}(\xi)t}(1+|\xi|^{2})^{j-k}}{(\lambda_{+}(\xi)-\lambda_{-}(\xi))(1+|\xi|^{2})^{j}}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]\mathrm{d}\xi.$

Since

$\displaystyle\left|\partial_{\xi}^{\eta}\left(\frac{\lambda_{-}(\xi)^{j}e^{\lambda_{-}(\xi)t}(1+|\xi|^{2})^{j-k}}{(\lambda_{+}(\xi)-\lambda_{-}(\xi))(1+|\xi|^{2})^{j}}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right)\right|\leq
C_{\eta}t^{-(j-k)}e^{-c_{1}t-c_{2}|\xi|^{2}t}|\xi|^{-|\eta|-2},$

we obtain

$\displaystyle\begin{array}[]{l}\displaystyle\left|\mathcal{F}^{-1}\left[\frac{\lambda_{-}(\xi)^{j}e^{\lambda_{-}(\xi)t}(1+|\xi|^{2})^{j-k}}{(\lambda_{+}(\xi)-\lambda_{-}(\xi))(1+|\xi|^{2})^{j}}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]\right|\\\\[6.45831pt]
\displaystyle\quad\leq
C_{m}|x|^{-m}t^{-(j-k)}e^{-c_{1}t}\int_{|\xi|\geq\sqrt{2}M_{2}}e^{-c_{2}|\xi|^{2}t}|\xi|^{-m-2}\mathrm{d}\xi\\\\[8.61108pt]
\displaystyle\quad\leq\begin{cases}C_{m}|x|^{-m}t^{-(j-k)-\frac{3}{2}}e^{-c_{1}t},&m\geq
0,\\\ C_{m}|x|^{-m}t^{-(j-k)}e^{-c_{1}t},&m\geq 2.\end{cases}\end{array}$
(5.21)

Therefore, by using $(5.9)$, we have

$\displaystyle\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\lambda_{-}}(t)Qf]\|_{L^{1}}$
$\displaystyle\leq\left\|\mathcal{F}^{-1}\left[\frac{\lambda_{-}(\xi)^{j}e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{\varphi}_{\infty}(\xi)\frac{{\xi}^{\top}\xi}{|\xi|^{2}}\right]\right\|_{L^{1}}\|\partial_{x}^{\alpha}(1-\Delta)^{k}f\|_{L^{1}}$
$\displaystyle\leq Ct^{-(j-k)}e^{-c_{1}t}\int_{|x|\leq
1}|x|^{-2}\mathrm{d}x\|f\|_{W^{2k+|\alpha|,1}}$
$\displaystyle\quad+Ct^{-(j-k)}e^{-c_{1}t}\int_{|x|\geq
1}|x|^{-4}\mathrm{d}x\|f\|_{W^{2k+|\alpha|,1}}$ $\displaystyle\leq
Ct^{-(j-k)}e^{-ct}\|f\|_{W^{2k+|\alpha|,1}}.$

Similarly, we obtain

$\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{\infty}^{\mu_{-}}(t)Qf]\right\|_{L^{1}}\leq
Ct^{-(j-k)}e^{-ct}\|f\|_{W^{2k+|\alpha|,1}}.$

This completes the proof of $(5.6)$. $\blacksquare$

We next investigate the middle frequency part.

######  Lemma 5.2.

The following estimates hold for $\alpha\in(\mathbb{N}\cup\\{0\\})^{3}$,
$j\geq 0$ and $t\geq 0$:

$\displaystyle(i)~{}\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{M}^{\lambda_{\pm}}(t)f]\|_{L^{1}}+\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{M}^{\mu_{\pm}}(t)f]\right\|_{L^{1}}\leq
Ce^{-ct}\|f\|_{L^{1}},$ (5.22)
$\displaystyle(ii)~{}\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{M}_{M}^{\lambda_{\pm}}(t)f]\|_{L^{1}}+\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{M}_{M}^{\mu_{\pm}}(t)f]\right\|_{L^{1}}\leq
Ce^{-ct}\|f\|_{L^{1}},$ (5.23)
$\displaystyle(iii)~{}\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{M}^{\lambda_{\pm}}(t)Qf]\|_{L^{1}}+\left\|\partial_{t}^{j}\partial_{x}^{\alpha}[\mathcal{K}_{M}^{\mu_{\pm}}(t)Qf]\right\|_{L^{1}}\leq
Ce^{-ct}\|f\|_{L^{1}}.$ (5.24)

Proof. The following formulas hold for
$\frac{M_{1}}{\sqrt{2}}\leq|\xi|\leq\sqrt{2}M_{2}$

$\frac{e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}=\frac{1}{2\pi
i}\int_{\Gamma}\frac{e^{zt}}{z^{2}+\nu|\xi|^{2}z+\beta^{2}|\xi|^{2}}\mathrm{d}z,$
$\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}=\frac{1}{2\pi
i}\int_{\Gamma}\frac{e^{zt}}{z^{2}+(\nu+\tilde{\nu})|\xi|^{2}z+(\beta^{2}+\gamma^{2})|\xi|^{2}}\mathrm{d}z.$

Here $\Gamma$ is a closed path containing $\lambda_{\pm}(\xi)$ and
$\mu_{\pm}(\xi)$, and included in $\\{z\in\mathbb{C}|\mathrm{Re}z\leq-
c_{3}\\}$. Here $c_{3}$ is a positive number taken by

$\max_{\frac{M_{1}}{\sqrt{2}}\leq|\xi|\leq\sqrt{2}M_{2}}\mathrm{Re}\mu_{j}(\xi)\leq-2c_{3},j=1,2,3,4.$

Hence, we can compute as in a similar manner to [11,20] to obtain

$\displaystyle\left|\partial_{t}\partial_{x}^{\alpha}\mathcal{F}^{-1}\left[\frac{1}{2\pi
i}\int_{\Gamma}\frac{e^{zt}}{z^{2}+\nu|\xi|^{2}z+\beta^{2}|\xi|^{2}}\mathrm{d}z\eta(\xi)\hat{\varphi}_{M}(\xi)\right]\right|$
$\displaystyle\leq C_{j,\alpha,N}e^{-ct}|x|^{-N},j+|\alpha|\geq 1,N\geq 0,$
$\displaystyle\left|\partial_{t}\partial_{x}^{\alpha}\mathcal{F}^{-1}\left[\frac{1}{2\pi
i}\int_{\Gamma}\frac{e^{zt}}{z^{2}+(\nu+\tilde{\nu})|\xi|^{2}z+(\beta^{2}+\gamma^{2})|\xi|^{2}}\mathrm{d}z\eta(\xi)\hat{\varphi}_{M}(\xi)\right]\right|$
$\displaystyle\leq C_{j,\alpha,N}e^{-ct}|x|^{-N},j+|\alpha|\geq 1,N\geq 0.$

Here $\eta$ is a function such that $\eta\in
C^{\infty}(S^{2}),~{}S^{2}=\\{\xi\in\mathbb{R}^{3}~{}|~{}|\xi|=1\\}$ and
$\eta=\eta(\frac{\xi}{|\xi|})$. Therefore we have $(5.10)$–$(5.12)$. This
completes the proof. $\blacksquare$
Proof of Proposition 4.3 (ii). By using Lemma 5.1 and Lemma 5.2, we have the
following estimates of $\mathcal{K}^{j1}(t)P_{\infty}\phi_{0},~{}j=1,2,$ and
$\mathcal{K}^{j2}(t)P_{\infty}w_{0},~{}j=1,2,3$

$\displaystyle\|\mathcal{K}^{11}(t)P_{\infty}\phi_{0}\|_{L^{1}}$
$\displaystyle\leq Ce^{-ct}\|\phi_{0}\|_{L^{1}},$
$\displaystyle\|\mathcal{K}^{12}(t)P_{\infty}w_{0}\|_{L^{1}}$
$\displaystyle\leq Ce^{-ct}\|w_{0}\|_{L^{1}},$
$\displaystyle\|\mathcal{K}^{21}(t)P_{\infty}\phi_{0}\|_{L^{1}}$
$\displaystyle\leq Ce^{-ct}\|\phi_{0}\|_{L^{1}},$
$\displaystyle\|\mathcal{K}^{22}(t)P_{\infty}w_{0}\|_{L^{1}}$
$\displaystyle\leq Ce^{-ct}\|w_{0}\|_{W^{2,1}},$
$\displaystyle\|\mathcal{K}^{32}(t)P_{\infty}w_{0}\|_{L^{1}}$
$\displaystyle\leq Ce^{-ct}\|w_{0}\|_{W^{2,1}},$

We next focus on $\mathcal{K}^{j3}(t)P_{\infty}G_{0},~{}j=2,3.$ By using
$G_{0}=\nabla\psi_{0}$, we write
$\mathcal{F}[\mathcal{K}^{j3}(t)G_{0}],~{}j=2,3$ as

$\displaystyle\mathcal{F}[\mathcal{K}^{23}(t)G_{0}](\xi)$
$\displaystyle=-i\beta^{2}\frac{e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}(\hat{G}_{0}(\xi)-\mathrm{tr}(\hat{G}_{0}(\xi))I)\xi\
$
$\displaystyle\quad\quad-i\beta^{2}\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}\mathrm{tr}(\hat{G}_{0}(\xi))\xi,$
$\displaystyle\mathcal{F}[\mathcal{K}^{33}(t)G_{0}](x)$
$\displaystyle=\frac{\lambda_{+}(\xi)e^{\lambda_{+}(\xi)t}-\lambda_{-}(\xi)e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{G}_{0}(\xi)+\nu|\xi|^{2}\frac{e^{\lambda_{-}(\xi)t}-e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\hat{G}_{0}(\xi)$
$\displaystyle+\left((\nu+\tilde{\nu})\frac{e^{\mu_{+}(\xi)t}-e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}-\nu\frac{e^{\lambda_{-}(\xi)t}-e^{\lambda_{+}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\right)\xi{}^{\top}\xi
G_{0}(\xi)$
$\displaystyle\quad\quad+\left(\frac{\mu_{+}(\xi)e^{\mu_{+}(\xi)t}-\mu_{-}(\xi)e^{\mu_{-}(\xi)t}}{\mu_{+}(\xi)-\mu_{-}(\xi)}-\frac{\lambda_{+}(\xi)e^{\lambda_{+}(\xi)t}-\lambda_{-}(\xi)e^{\lambda_{-}(\xi)t}}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}\right)\frac{\xi^{\top}\xi}{|\xi|^{2}}\hat{G}_{0}(\xi).$

It then follows from Lemma 5.1 and Lemma 5.2 that

$\displaystyle\|\mathcal{K}^{23}(t)P_{\infty}G_{0}\|_{L^{1}}$
$\displaystyle\leq Ce^{-ct}\|G_{0}\|_{W^{2,1}},$
$\displaystyle\|\mathcal{K}^{33}(t)P_{\infty}G_{0}\|_{L^{1}}$
$\displaystyle\leq Ce^{-ct}\|G_{0}\|_{W^{2,1}}.$

Consequently we arrive at

$\|e^{-tL}P_{\infty}U_{0}\|_{L^{1}}\leq Ce^{-ct}\|U_{0}\|_{W^{2,1}}.$

This completes the proof. $\blacksquare$

Acknowledgements.  This work was partially supported by JSPS KAKENHI Grant
Number 19J10056.

## References

  * [1] M. E. Gurtin,  An Introduction to Continuum Mechanics,  Math. Sci. Eng., vol. 158, Academic Press, New York–London, 1981.
  * [2] D. Hoff, K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., $\bf{44}$ (1995), pp. 603–676.
  * [3] X. Hu, Global existence of weak solutions to two dimensional compressible viscoelastic flows, J. Differential Equations, $\bf{265}$ (2018), pp. 3130–3167.
  * [4] X. Hu, D. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, $\bf{249}$ (2010), pp. 1179–1198.
  * [5] X. Hu, D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, $\bf{250}$ (2011), pp. 1200–1231.
  * [6] X. Hu, G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., $\bf{45}$ (2013), pp. 2815–2833.
  * [7] X. Hu, W. Zhao, Global existence of compressible dissipative elastodynamics systems with zero shear viscosity in two dimensions, Arch. Ration. Mech. Anal., $\bf{235}$ (2020), pp. 1177–1243.
  * [8] X. Hu, W. Zhao, Global existence for the compressible viscoelastic system with zero shear viscosity in three dimensions, J. Differential Equations, $\bf{268}$ (2020), pp. 1658–1685.
  * [9] Y. Ishigaki, Diffusion wave phenomena and $L^{p}$ decay estimates of solutions of compressible viscoelastic system, J. Differential Equations, $\bf{269}$ (2020), pp. 11195–11230.
  * [10] S. Kawashima, A. Matsumura, T. Nishida, On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Commun. Math. Phys., $\bf{70}$, (1979), pp.97–124.
  * [11] T. Kobayashi, Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equation, Pacific J. Math. $\bf{207}$ (2002), pp. 199–234.
  * [12] Y. Li, R. Wei, Z. Yao, Optimal decay rates for the compressible viscoelastic flows, J. Math. Phys., $\bf{57}$, 111506, (2016).
  * [13] F. Lin, C. Liu, P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., $\bf{58}$ (2005), pp. 1437–1471.
  * [14] L. Liu, Y. Wu, F. Xu, X. Zhang, The optimal convergence rates for the multi-dimensional compressible viscoelastic flows, Z. Angew. Math. Mech. $\bf{96}$, (2016), pp.1490–-1504.
  * [15] A. Matsumura, T. Nishida, The initial value problems for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser., 89 (1979), pp. 337–342.
  * [16] A. Matsumura, T. Nishida, Initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), pp. 67–104.
  * [17] X. Pan, J. Xu, Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^{p}$ critical spaces, Discrete Contin. Dyn. Syst., $\bf{39}$, (2019), pp. 2021–2057.
  * [18] J. Qian, Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., $\bf{198}$ (2010), pp. 835–868.
  * [19] J. Qian, Initial boundary value problems for the compressible viscoelastic fluid, J. Differential Equations, $\bf{250}$ (2011), pp. 848–865.
  * [20] Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., $\bf{23}$ (2000), pp. 203–226.
  * [21] T. C. Sideris, B. Thomases, Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., $\bf{57}$ (2004), pp. 1–39.
  * [22] R. Wei, Y. Li, Z. Yao, Decay of the compressible viscoelastic flows, Commun. Pure Appl. Anal., $\bf{15}$ (2016), pp. 1603–-1624.
  * [23] G. Wu, Z. Gao, Z. Tan, Time decay rates for the compressible viscoelastic flows, J. Math. Anal. Appl., $\bf{452}$ (2017), pp. 990–-1004.


# Electrostatic shock structures in a magnetized plasma having non-thermal
particles

S. Jahan∗,1, S. Banik∗∗,1,2, N.A. Chowdhury∗∗∗,3, A. Mannan†,1, and A.A.
Mamun§,1 1Department of Physics, Jahangirnagar University, Savar, Dhaka-1342,
Bangladesh
2Health Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh
3Plasma Physics Division, Atomic Energy Centre, Dhaka-1000, Bangladesh
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A rigorous theoretical investigation has been made on the nonlinear
propagation of dust-ion-acoustic shock waves in a multi-component magnetized
pair-ion plasma having inertial warm positive and negative ions, inertialess
non-thermal electrons and positrons, and static negatively charged massive
dust grains. The Burgers’ equation is derived by employing reductive
perturbation method. The plasma model supports both positive and negative
shock structures in the presence of static negatively charged massive dust
grains. It is found that the steepness of both positive and negative shock
profiles declines with the increase of ion kinematic viscosity without
affecting the height, and the temperature of the electrons enhances the
amplitude of the shock profile. It is also observed that the increase in
oblique angle rises the height of the positive shock profile, and the height
of the positive shock wave increases with the number density of positron. The
application of the findings from present investigation are briefly discussed.

###### keywords:

Pair-ion , Magnetized plasma , Ion-acoustic waves , Reductive perturbation
method , Shock waves.

††journal: “Plasma Physics Reports”

##  1 Introduction

Positive ions are produced by the electron impact ionization while negative
ions are produced due to the attachment of electron with an atom [1], and the
existence of both positive and negative ions or pair-ion (PI) can be observed
in space plasmas, viz., cometary comae [2], upper regions of Titan’s
atmosphere [3, 4, 5], plasmas in the D and F-regions of Earth’s ionosphere [4,
5, 6]) and also laboratory plasmas, viz., ($K^{+}$, $SF_{6}^{-}$) plasma [7,
8], ($Ar^{+}$, $F^{-}$) plasma [9], plasma processing reactors [10], plasma
etching [11], combustion products [11], ($Xe^{+}$, $F^{-}$) plasma [12],
neutral beam sources [13], ($Ar^{+}$, $SF_{6}^{-}$) plasma [14, 15,
16],($Ar^{+}$, $O_{2}^{-}$) plasma, and Fullerene ($C_{60}^{+}$, $C_{60}^{-}$)
plasma [17, 18], etc. The dynamics of the plasma system and associated
nonlinear electrostatic structures have been rigorously changed by the
presence of massive dust grains in the PI plasma (PIP) [19, 20, 21, 22].
Yasmin et al. [23] studied the nonlinear propagation of dust-ion-acoustic
(DIA) waves (DIAWs) in a multi-component plasma, and found that the shock
profile associated with DIAWs is significantly modified by the existence of
dust grains. A number of authors also examined the effects of the positron to
the formation of solitary profile associated with electrostatic waves [24,
25]. Rahman et al. [24] studied the electrostatic solitary waves in electron-
positron-ion plasma, and observed that the amplitude of the solitary profile
increases with increasing the number density of positron. Abdelsalam [25]
investigated ion-acoustic (IA) solitary waves in a dense plasma, and
demonstrated that the presence of the positron can cause to increase the
amplitude of the solitary profile.

Cairns et al. [26] first demonstrated the non-thermal distribution to
investigate the effect of energetic particles on the formation of IA shock
profile, and introduced the parameter $\alpha$ in the non-thermal distribution
for measuring the amount of deviation of non-thermal plasma species from
Maxellian-Boltzmann distribution [27]. The non-thermal plasma species are
regularly seen in the comtary comae [2], Earth’s ionosphere [4], and the upper
region of the Titans [3], etc. Haider et al. [28] investigated the IA solitary
waves in the presence of non-thermal particles, and observed that the width of
the solitary profile decreases with increasing of ions non-thermality. Pakzad
and Javidan [29] studied the dust-acoustic (DA) solitary and shock waves in a
dusty plasma having non-thermal ions, and reported that the amplitude of the
wave increases with the decrease of the non-thermality of ions. Ghai et al.
[30] studied the DA solitary waves in the presence of non-thermal ions, and
found that the height of the solitary wave decreases with the increase of
$\alpha$.

Landau damping and the kinematic viscosity among the plasma species are the
primary reason to the formation of shock profile associated with electrostatic
waves [31, 32, 33]. The existence of the external magnetic field is considered
to be responsible to change the configuration of the shock profile. Sabetkar
and Dorranian [31] examined the effects of external magnetic field to the
formation of the DA solitary waves in the presence of non-thermal plasma
species, and found that the amplitude of solitary wave increases with the
increase in the value of oblique angle. Shahmansouri and Mamun [32] analysis
the DA shock waves in a magnetized non-thermal dusty plasma and demonstrated
that the amplitude of shock wave increases with increasing the oblique angle.
Malik et al. [33] studied the small amplitude DA wave in magnetized plasma,
and reported that the height of the shock wave enhances with oblique angle.
Bedi et al. [34] studied DA solitary waves in a four-component magnetized
dusty plasma, and highlighted that both compressive and rarefactive solitons
can exist in the presence of external magnetic field. To the best knowledge of
the authors, no attempt has been made to study the DIA shock waves (DIASHWs)
in a magnetized PIP by considering kinematic viscosity of both inertial warm
positive and negative ion species, and inertialess non-thermal electrons and
positrons in the presence of static negatively charged dust grains. The aim of
our present investigation is, therefore, to derive Burgers’ equation and
investigate DIASHWs in a magnetized PIP, and to observe the effects of various
plasma parameters (e.g., mass, charge, temperature, kinematic viscosity, and
obliqueness, etc.) on the configuration of DIASHWs.

The layout of the paper is as follows: The basic equations are displayed in
section 2. The well known Burgers’ equation has been derived in section 3.
Numerical analysis and discussion are presented in section 4. A brief
conclusion is pinpointed in section 5.

##  2 Governing equations

We consider a multi-component PIP having inertial positively charged warm
ions, (mass $m_{1}$; charge $eZ_{1}$; temperature $T_{1}$; number density
$\tilde{n_{1}}$), negatively charged warm ions (mass $m_{2}$; charge
$-eZ_{2}$; temperature $T_{2}$; number density $\tilde{n_{2}}$), inertialess
electrons, featuring non-thermal distribution (mass $m_{e}$; charge $-e$;
temperature $T_{e}$; number density $\tilde{n_{e}}$), inertialess positrons,
obeying non-thermal distribution (mass $m_{p}$; charge $e$; temperature
$T_{p}$; number density $\tilde{n_{p}}$) and static negatively charged massive
dust grains (charge $-eZ_{d}$; number density $n_{d}$); where $Z_{1}$
($Z_{2}$) is the charge state of the positive (negative) ion, and $Z_{d}$ is
the charge state of the negative dust grains, and $e$ is the magnitude of the
charge of an electron. An external magnetic field $\mathbf{B}_{0}$ has been
considered in the system directed along the $z$-axis defining
$\mathbf{B}_{0}=B_{0}\hat{z}$, where $B_{0}$ and $\hat{z}$ denoted the
strength of the external magnetic field and unit vector directed along the
$z$-axis, respectively. The dynamics of the PIP system is governed by the
following set of equations [35, 36, 37, 38]

$\displaystyle\frac{\partial\tilde{n}_{1}}{\partial\tilde{t}}+\acute{\nabla}\cdot(\tilde{n}_{1}\tilde{u}_{1})=0,$
(1)
$\displaystyle\frac{\partial\tilde{u}_{1}}{\partial\tilde{t}}+(\tilde{u}_{1}\cdot\acute{\nabla})\tilde{u}_{1}=-\frac{Z_{1}e}{m_{1}}\acute{\nabla}\tilde{\psi}+\frac{Z_{1}eB_{0}}{m_{1}}(\tilde{u}_{1}\times\hat{z})$
$\displaystyle\hskip
71.13188pt-\frac{1}{m_{1}\tilde{n}_{1}}\acute{\nabla}P_{1}+\tilde{\eta}_{1}\acute{\nabla}^{2}\tilde{u}_{1},$
(2)
$\displaystyle\frac{\partial\tilde{n}_{2}}{\partial\tilde{t}}+\acute{\nabla}\cdot(\tilde{n}_{2}\tilde{u}_{2})=0,$
(3)
$\displaystyle\frac{\partial\tilde{u}_{2}}{\partial\tilde{t}}+(\tilde{u}_{2}\cdot\acute{\nabla})\tilde{u}_{2}=\frac{Z_{2}e}{m_{2}}\acute{\nabla}\tilde{\psi}-\frac{Z_{2}eB_{0}}{m_{2}}(\tilde{u}_{2}\times\hat{z})$
$\displaystyle\hskip
71.13188pt-\frac{1}{m_{2}\tilde{n}_{2}}\acute{\nabla}P_{2}+\tilde{\eta}_{2}\acute{\nabla}^{2}\tilde{u}_{2},$
(4) $\displaystyle\acute{\nabla}^{2}\tilde{\psi}=4\pi
e[\tilde{n}_{e}+Z_{d}\tilde{n}_{d}+Z_{2}\tilde{n}_{2}-Z_{1}\tilde{n}_{1}-\tilde{n}_{p}],\
$ (5)

where $\tilde{u}_{2}$ ($\tilde{u}_{2}$) is the positive (negative) ion fluid
velocity; $\tilde{\eta}_{1}=\mu_{1}/m_{1}n_{1}$
($\tilde{\eta}_{2}=\mu_{2}/m_{2}n_{2}$) is the kinematic viscosity of the
positive (negative) ion; $P_{1}$ ($P_{2}$) is the pressure of positive
(negative) ion, and $\tilde{\psi}$ represents the electrostatic wave
potential. Now, we are introducing normalized variables, namely,
$n_{1}\rightarrow\tilde{n}_{1}/n_{10}$,
$n_{2}\rightarrow\tilde{n}_{2}/n_{20}$,
$n_{e}\rightarrow\tilde{n}_{e}/n_{e0}$,
$n_{p}\rightarrow\tilde{n}_{p}/n_{p0}$,
$n_{d}\rightarrow\tilde{n}_{d}/n_{d0}$, $u_{1}\rightarrow\tilde{u}_{1}/C_{2}$,
$u_{2}\rightarrow\tilde{u}_{2}/C_{2}$ [where
$C_{2}=(Z_{2}k_{B}T_{e}/m_{2})^{1/2}$, $k_{B}$ being the Boltzmann constant];
$\psi\rightarrow\tilde{\psi}e/k_{B}T_{e}$; $t=\tilde{t}/\omega_{P_{2}}^{-1}$
[where $\omega_{P_{2}}^{-1}=(m_{2}/4\pi e^{2}Z_{2}^{2}n_{20})^{1/2}$];
$\nabla=\acute{\nabla}/\lambda_{D}$ [where $\lambda_{D}=(k_{B}T_{e}/4\pi
e^{2}Z_{2}n_{20})^{1/2}$]. The pressure term of the positive ion can be
recognized as $P_{1}=P_{10}(N_{1}/n_{10})^{\gamma}$ with
$P_{10}=n_{10}k_{B}T_{1}$ being the equilibrium pressure of the positive ion,
and the pressure term of the negative ion can be recognized as
$P_{2}=P_{20}(N_{2}/n_{20})^{\gamma}$ with $P_{20}=n_{20}k_{B}T_{2}$ being the
equilibrium pressure of the negative ion, and $\gamma=(N+2)/N$ (where $N$ is
the degree of freedom and for three-dimensional case $N=3$, then
$\gamma=5/3$). For simplicity, we have considered
($\tilde{\eta}_{1}\approx\tilde{\eta}_{2}=\eta$), and $\eta$ is normalized by
$\omega_{p_{2}}\lambda_{D}^{2}$. The quasi-neutrality condition at equilibrium
for our plasma model can be written as $Z_{1}n_{10}+n_{p0}\approx
Z_{2}n_{20}+Z_{d}n_{d0}+n_{e0}$. Equations (1)$-$(5) can be expressed in the
normalized form as [39]:

$\displaystyle\frac{\partial n_{1}}{\partial t}+\nabla\cdot(n_{1}u_{1})=0,$
(6) $\displaystyle\frac{\partial u_{1}}{\partial
t}+(u_{1}\cdot\nabla)u_{1}=-\alpha_{1}\nabla\psi+\alpha_{1}\Omega_{c}(u_{1}\times\hat{z})$
$\displaystyle\hskip 71.13188pt-\alpha_{2}\nabla
n_{1}^{\gamma-1}+\eta\nabla^{2}u_{1},$ (7) $\displaystyle\frac{\partial
n_{2}}{\partial t}+\nabla\cdot(n_{2}u_{2})=0,$ (8)
$\displaystyle\frac{\partial u_{2}}{\partial
t}+(u_{2}\cdot\nabla)u_{2}=\nabla\psi-\Omega_{c}(u_{2}\times\hat{z})$
$\displaystyle\hskip 71.13188pt-\alpha_{3}\nabla
n_{2}^{\gamma-1}+\eta\nabla^{2}u_{2},$ (9)
$\displaystyle\nabla^{2}\psi=\lambda_{e}n_{e}-\lambda_{p}n_{p}+\lambda_{d}n_{d}-(1+\lambda_{e}+\lambda_{d}-\lambda_{p})n_{1}+n_{2}.\
$ (10)

Other plasma parameters can be recognized as
$\alpha_{1}=Z_{1}m_{2}/Z_{2}m_{1}$, $\alpha_{2}=5T_{1}m_{2}/2Z_{2}T_{e}m_{1}$,
$\alpha_{3}=5T_{2}/2Z_{2}T_{e}$, $\lambda_{e}=n_{e0}/Z_{2}n_{20}$,
$\lambda_{d}=Z_{d}n_{d0}/Z_{2}n_{20}$, $\lambda_{p}=n_{p0}/Z_{2}n_{20}$, and
$\Omega_{c}=\omega_{c}/\omega_{p_{2}}$ [where $\omega_{c}=Z_{2}eB_{0}/m_{2}$].
Now, the expression for the number density of electrons and positrons
following non-thermal distribution can be, respectively, written as [26]

$\displaystyle n_{e}=(1-\beta\psi+\beta\psi^{2})\mbox{exp}(\psi),$ (11)
$\displaystyle
n_{p}=(1+\beta\alpha_{4}\psi+\beta\alpha_{4}^{2}\psi^{2})\mbox{exp}(-\alpha_{4}\psi),\
$ (12)

where $\beta=4\alpha/(1+3\alpha)$, ($\alpha$ represents the number of non-
thermal populations in our considered model) and $\alpha_{4}=T_{e}/T_{p}$.
Now, by substituting Eqs. (11)-(12) into the Eq. (10), and expanding up to
third order in $\psi$, we can write

$\displaystyle\nabla^{2}\psi=\lambda_{e}-\lambda_{p}+n_{d}\lambda_{d}+n_{2}-\Lambda
n_{1}+\sigma_{1}\psi+\sigma_{2}\psi^{2}+\cdot\cdot\cdot,\ $ (13)

where $\Lambda=1+\lambda_{e}+\lambda_{d}-\lambda_{p}$,
$\sigma_{1}=\lambda_{e}(1-\beta)-\lambda_{p}\alpha_{4}(\beta-1)$, and
$\sigma_{2}=\lambda_{e}/2-\lambda_{p}\alpha_{4}^{2}/2$. We note that the terms
containing $\sigma_{1}$ and $\sigma_{2}$ are the contribution of non-thermal
distributed electrons and positrons.

##  3 Derivation of the Burgers’ equation

To derive the Burgers’ equation for the DIASHWs propagating in a PIP, we first
introduce the stretched co-ordinates [40, 41] as

$\displaystyle\xi=\epsilon(l_{x}x+l_{y}y+l_{z}z-V_{p}t),$ (14)
$\displaystyle\tau={\epsilon}^{2}t,\ $ (15)

where $V_{p}$ is the phase speed and $\epsilon$ is a smallness parameter
denoting the weakness of the dissipation ($0<\epsilon<1$). It is noted that
$l_{x}$, $l_{y}$, and $l_{z}$ (i.e., $l_{x}^{2}+l_{y}^{2}+l_{z}^{2}=1$) are
the directional cosines of the wave vector of $k$ along $x$, $y$, and
$z$-axes, respectively. Then, the dependent variables can be expressed in
power series of $\epsilon$ as [41]

$\displaystyle n_{1}=1+\epsilon
n_{1}^{(1)}+\epsilon^{2}n_{1}^{(2)}+\epsilon^{3}n_{1}^{(3)}+\cdot\cdot\cdot,$
(16) $\displaystyle n_{2}=1+\epsilon
n_{2}^{(1)}+\epsilon^{2}n_{2}^{(2)}+\epsilon^{3}n_{2}^{(3)}+\cdot\cdot\cdot,$
(17) $\displaystyle
u_{1x,y}=\epsilon^{2}u_{1x,y}^{(1)}+\epsilon^{3}u_{1x,y}^{(2)}+\cdot\cdot\cdot,$
(18) $\displaystyle
u_{2x,y}=\epsilon^{2}u_{2x,y}^{(1)}+\epsilon^{3}u_{2x,y}^{(2)}+\cdot\cdot\cdot,$
(19) $\displaystyle u_{1z}=\epsilon
u_{1z}^{(1)}+\epsilon^{2}u_{1z}^{(2)}+\cdot\cdot\cdot,$ (20) $\displaystyle
u_{2z}=\epsilon u_{2z}^{(1)}+\epsilon^{2}u_{2z}^{(2)}+\cdot\cdot\cdot,$ (21)
$\displaystyle\psi=\epsilon\psi^{(1)}+\epsilon^{2}\psi^{(2)}+\cdot\cdot\cdot.\
$ (22)

Now, by substituting Eqs. (14)$-$(22) into Eqs. (6)$-$(9), and (13), and
collecting the terms containing $\epsilon$, the first-order equations become

$\displaystyle
n_{1}^{(1)}=\frac{3\alpha_{1}l_{z}^{2}}{3V_{p}^{2}-2\alpha_{2}l_{z}^{2}}\psi^{(1)},$
(23) $\displaystyle
u_{1z}^{(1)}=\frac{3V_{p}\alpha_{1}l_{z}}{3V_{p}^{2}-2\alpha_{2}l_{z}^{2}}\psi^{(1)},$
(24) $\displaystyle
n_{2}^{(1)}=-\frac{3l_{z}^{2}}{3V_{p}^{2}-2\alpha_{3}l_{z}^{2}}\psi^{(1)},$
(25) $\displaystyle
u_{2z}^{(1)}=-\frac{3V_{p}l_{z}}{3V_{p}^{2}-2\alpha_{3}l_{z}^{2}}\psi^{(1)}.\
$ (26)

Now, the phase speed of DIASHWs can be read as

$\displaystyle V_{p}\equiv
V_{p+}=l_{z}\sqrt{{\frac{-a_{1}+\sqrt{a_{1}^{2}-36\sigma_{1}a_{2}}}{18\sigma_{1}}}},$
(27) $\displaystyle V_{p}\equiv
V_{p-}=l_{z}\sqrt{{\frac{-a_{1}-\sqrt{a_{1}^{2}-36\sigma_{1}a_{2}}}{18\sigma_{1}}}},\
$ (28)

where $a_{1}=-{6\sigma_{1}\alpha_{2}+6\sigma_{1}\alpha_{3}+9+\Lambda n_{1}}$
and
$a_{2}=4\sigma_{1}\alpha_{2}\alpha_{3}+6\alpha_{2}+6\alpha_{1}\alpha_{3}\Lambda$.
The $x$ and $y$-components of the first-order momentum equations can be
written as

$\displaystyle
u_{1x}^{(1)}=-\frac{3l_{y}V_{p}^{2}}{\Omega_{c}(3V_{p}^{2}-2\alpha_{2}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi},$
(29) $\displaystyle
u_{1y}^{(1)}=\frac{3l_{x}V_{p}^{2}}{\Omega_{c}(3V_{p}^{2}-2\alpha_{2}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi},$
(30) $\displaystyle
u_{2x}^{(1)}=-\frac{3l_{y}V_{p}^{2}}{\Omega_{c}(3V_{p}^{2}-2\alpha_{3}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi},$
(31) $\displaystyle
u_{2y}^{(1)}=\frac{3l_{x}V_{p}^{2}}{\Omega_{c}(3V_{p}^{2}-2\sigma_{3}l_{z}^{2})}~{}\frac{\partial\psi^{(1)}}{\partial\xi}.\
$ (32)

Now, by following the next higher-order terms, the equation of continuity,
momentum equation, and Poisson’s equation can be written as

$\displaystyle\frac{\partial n_{1}^{(1)}}{\partial\tau}-V_{p}\frac{\partial
n_{1}^{(2)}}{\partial\xi}+l_{x}\frac{\partial
u_{1x}^{(1)}}{\partial\xi}+l_{y}\frac{\partial u_{1y}^{(1)}}{\partial\xi}$
$\displaystyle\hskip 42.67912pt+l_{z}\frac{\partial
u_{1z}^{(2)}}{\partial\xi}+l_{z}\frac{\partial}{\partial\xi}\big{(}n_{1}^{(1)}u_{+z}^{(1)}\big{)}=0,$
(33) $\displaystyle\frac{\partial
u_{1z}^{(1)}}{\partial\tau}-V_{p}\frac{\partial
u_{1z}^{(2)}}{\partial\xi}+l_{z}u_{1z}^{(1)}\frac{\partial
u_{1z}^{(1)}}{\partial\xi}+\alpha_{1}l_{z}\frac{\partial\psi^{(2)}}{\partial\xi}$
$\displaystyle\hskip
14.22636pt+\alpha_{2}l_{z}\frac{\partial}{\partial\xi}\bigg{[}\frac{2}{3}n_{1}^{(2)}-\frac{1}{9}(n_{1}^{(1)})^{2}\bigg{]}-\eta\frac{\partial^{2}u_{1z}^{(1)}}{\partial\xi^{2}}=0,$
(34) $\displaystyle\frac{\partial
n_{2}^{(1)}}{\partial\tau}-V_{p}\frac{\partial
n_{2}^{(2)}}{\partial\xi}+l_{x}\frac{\partial
u_{2x}^{(1)}}{\partial\xi}+l_{y}\frac{\partial u_{2y}^{(1)}}{\partial\xi}$
$\displaystyle\hskip 42.67912pt+l_{z}\frac{\partial
u_{2z}^{(2)}}{\partial\xi}+l_{z}\frac{\partial}{\partial\xi}\big{(}n_{2}^{(1)}u_{2z}^{(1)}\big{)}=0,$
(35) $\displaystyle\frac{\partial
u_{2z}^{(1)}}{\partial\tau}-V_{p}\frac{\partial
u_{2z}^{(2)}}{\partial\xi}+l_{z}u_{2z}^{(1)}\frac{\partial
u_{2z}^{(1)}}{\partial\xi}-l_{z}\frac{\partial\psi^{(2)}}{\partial\xi}$
$\displaystyle\hskip
14.22636pt+\alpha_{3}l_{z}\frac{\partial}{\partial\xi}\bigg{[}\frac{2}{3}n_{2}^{(2)}-\frac{1}{9}(n_{2}^{(1)})^{2}\bigg{]}-\eta\frac{\partial^{2}u_{2z}^{(1)}}{\partial\xi^{2}}=0,$
(36)
$\displaystyle\sigma_{1}\psi^{(2)}+\sigma_{2}{[\psi^{(1)}]}^{2}+n_{2}^{(2)}-\Lambda
n_{1}^{(2)}=0.\ $ (37)

Finally, the next higher-order terms of Eqs. (6)$-$(9), and (12), with the
help of Eqs. (23)$-$(37), can provide the Burgers’ equation as

$\displaystyle\frac{\partial\Psi}{\partial\tau}+A\Psi\frac{\partial\Psi}{\partial\xi}=C\frac{\partial^{2}\Psi}{\partial\xi^{2}},\
$ (38)

where $\Psi=\psi^{(1)}$ is used for simplicity. In Eq. (38), the nonlinear
coefficient ($A$) and dissipative coefficient ($C$) are given by the following
expression

$\displaystyle
A=\frac{(M_{1}S_{1}^{3}-M_{2}S_{2}^{3}-2\sigma_{2}S_{1}^{3}S_{2}^{3})}{M_{3}S_{1}S_{2}},~{}~{}\mbox{and}~{}~{}C=\frac{\eta}{2},\
$ (39)

where
$M_{1}=\Lambda(81\alpha_{1}^{2}V_{p}^{2}l_{z}^{4}-6\alpha_{2}\alpha_{1}^{2}l_{z}^{6})$,
$M_{2}=81V_{p}^{2}l_{z}^{4}-6\alpha_{3}l_{z}^{6}$, and
$M_{3}=18V_{p}l_{z}^{2}{1+\alpha_{1}\Lambda}$. Now, we look forward to the
stationary shock wave solution of this Burgers’ equation by taking $\zeta=\xi-
U_{0}\tau^{\prime}$ and $\tau=\tau^{\prime}$, where $U_{0}$ is the speed of
the shock waves in the reference frame. These allow us to represent the
stationary shock wave solution as [42, 43, 41]

$\displaystyle\Psi=\Psi_{m}\Big{[}1-\tanh\bigg{(}\frac{\zeta}{\Delta}\bigg{)}\Big{]},\
$ (40)

where $\Psi_{m}$ is the amplitude and $\Delta$ is the width. The expression of
the amplitude and width can be given by the following equations

$\displaystyle\Psi_{m}=\frac{U_{0}}{A},~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}\Delta=\frac{2C}{U_{0}}.\
$ (41)

Figure 1: The variation of nonlinear coefficient $A$ with $\alpha_{4}$ along
with $\alpha_{1}=1.5$, $\alpha_{2}=0.05$, $\alpha_{3}=0.03$,
$\lambda_{p}=1.5$, $\lambda_{e}=1.7$, $\lambda_{d}=0.05$, $\delta=30^{\circ}$,
$\alpha=0.5$, and $V_{p}\equiv V_{p+}$. Figure 2: The variation of $\Psi$ with
$\zeta$ for different values of $\eta$ under consideration of
$\alpha_{4}>\alpha_{4c}$ along with $\alpha_{1}=1.5$, $\alpha_{2}=0.05$,
$\alpha_{3}=0.03$, $\alpha_{4}=1.5$, $\lambda_{p}=1.5$, $\lambda_{e}=1.7$,
$\lambda_{d}=0.05$, $\delta=30^{\circ}$, $\alpha=0.5$, $U_{0}=0.01$, and
$V_{p}\equiv V_{p+}$. Figure 3: The variation of $\Psi$ with $\zeta$ for
different values of $\eta$ under consideration of $\alpha_{4}<\alpha_{4c}$
along with $\alpha_{1}=1.5$, $\alpha_{2}=0.05$, $\alpha_{3}=0.03$,
$\alpha_{4}=0.5$, $\lambda_{p}=1.5$, $\lambda_{e}=1.7$, $\lambda_{d}=0.05$,
$\delta=30^{\circ}$, $\alpha=0.5$, $U_{0}=0.01$, and $V_{p}\equiv V_{p+}$.
Figure 4: The variation of $\Psi$ with $\zeta$ for different values of
$\delta$ under consideration of $\alpha_{4}>\alpha_{4c}$ along with
$\alpha_{1}=1.5$, $\alpha_{2}=0.05$, $\alpha_{3}=0.03$, $\alpha_{4}=1.5$,
$\lambda_{p}=1.5$, $\lambda_{e}=1.7$, $\lambda_{d}=0.05$, $\eta=0.3$,
$\alpha=0.5$, $U_{0}=0.01$, and $V_{p}\equiv V_{p+}$. Figure 5: The variation
of $\Psi$ with $\zeta$ for different values of $\alpha$ under consideration of
$\alpha_{4}>\alpha_{4c}$ along with $\alpha_{1}=1.5$, $\alpha_{2}=0.05$,
$\alpha_{3}=0.03$, $\alpha_{4}=1.5$, $\lambda_{p}=1.5$, $\lambda_{e}=1.7$,
$\lambda_{d}=0.05$, $\delta=30^{\circ}$, $\eta=0.3$, $U_{0}=0.01$, and
$V_{p}\equiv V_{p+}$. Figure 6: The variation of $\Psi$ with $\zeta$ for
different values of $\lambda_{p}$ under consideration of
$\alpha_{4}>\alpha_{4c}$ along with $\alpha_{1}=1.5$, $\alpha_{2}=0.05$,
$\alpha_{3}=0.03$, $\alpha_{4}=1.5$, $\lambda_{e}=1.7$, $\lambda_{d}=0.05$,
$\delta=30^{\circ}$, $\eta=0.3$, $\alpha=0.5$, $U_{0}=0.01$, and $V_{p}\equiv
V_{p+}$.

##  4 Numerical Analysis and Discussion

Now, we would like to observe the basic properties of DIASHWs in a magnetized
PIP having inertial pair-ions, inertialess non-thermal distributed electrons
and positrons, and static negatively charged massive dust grains by changing
the various plasma parameters, viz., ion kinematic viscosity, oblique angle,
non-thermality of electrons and positrons, mass, charge, temperature, and
number density of the plasma species. Equation (41) shows that under
consideration $U_{0}>0$ and $C>0$, no shock wave will exist if $A=0$ as the
amplitude of the wave becomes infinite which clearly violates the reductive
perturbation method. So, $A$ can be positive (i.e., $A>0$) or negative (i.e.,
$A<0$) according to the value of other plasma parameters. Figure 1 illustrates
the variation of $A$ with $\alpha_{4}$, and it is obvious from this figure
that $A$ can be negative, zero, and positive according to the values of
$\alpha_{4}$ when other plasma parameters are $\alpha_{1}=1.5$,
$\alpha_{2}=0.05$, $\alpha_{3}=0.03$, $\lambda_{p}=1.5$, $\lambda_{e}=1.7$,
$\lambda_{d}=0.05$, $\delta=30^{\circ}$, and $\alpha=0.5$. The point at which
$A$ becomes zero for the value of $\alpha_{4}$ is known as the critical value
of $\alpha_{4}$ (i.e., $\alpha_{4c}$). In our present analysis, the critical
value of $\alpha_{4}$ is $\alpha_{4c}\equiv 01$. So, the negative (positive)
shock profile can be exist for the value of $\alpha_{4}<\alpha_{4c}$
($\alpha_{4}>\alpha_{4c}$).

Figure 7: The variation of $\Psi$ with $\zeta$ for different values of
$\alpha_{1}$ under consideration of $\alpha_{4}>\alpha_{4c}$ along with
$\alpha_{2}=0.05$, $\alpha_{3}=0.03$, $\alpha_{4}=1.5$, $\lambda_{e}=1.7$,
$\lambda_{p}=1.5$, $\lambda_{d}=0.05$, $\delta=30^{\circ}$, $\eta=0.3$,
$\alpha=0.5$, $U_{0}=0.01$, and $V_{p}\equiv V_{p+}$.

Figures 2 and 3 respectively represent the variation of the positive and
negative shock profiles with ion kinematic viscosity (via $\eta$) when other
plasma parameters are remain constant. It is really interestingly that the
steepness of both positive and negative shock profiles declines with the
increase of $\eta$ without affecting the height. Figure 4 describes the
effects of the external magnetic field to the formation of the positive shock
profile. The increase in oblique angle rises the height of the positive shock
profile and this result is analogous to the result of Ref. [32].

The height of the positive shock profile is so much sensitive to the change of
non-thermality of the electrons and positrons which can be seen in Fig. 5.
There is a decrease in the amplitude of positive shock profile when electrons
and positrons deviate from thermodynamic equilibrium, and this result is
compatible with the result of Ref. [44]. The variation of the DIASHWs with
negative ion charge state, negative ion and positron number densities (via
$\lambda_{p}$) can be observed in Fig. 6. It is clear from Fig. 6 that as we
increase the positron (negative ion) number density, the height of the
positive shock wave increases (decreases) when the charge of the negative ion
remains constant or the the height of the positive shock wave decreases with
the charge of the negative ion for a fixed value of the number density of
positron and negative ion.

Figure 8: The variation of $\Psi$ with $\zeta$ for different values of
$\alpha_{2}$ under consideration of $\alpha_{4}>\alpha_{4c}$ along with
$\alpha_{1}=1.5$, $\alpha_{3}=0.03$, $\alpha_{4}=1.5$, $\lambda_{e}=1.7$,
$\lambda_{p}=1.5$, $\lambda_{d}=0.05$, $\delta=30^{\circ}$, $\eta=0.3$,
$\alpha=0.5$, $U_{0}=0.01$, and $V_{p}\equiv V_{p+}$.

The charge and mass of the positive and negative ions are rigourously
responsible to change the height of the positive shock profile. The variation
of the DIASHWs with $\alpha_{1}$ has been demonstrated in Fig. 7, and it is
obvious from this figure that the height of the positive shock profile
increases (decreases) with increasing the value of positive (negative) ion
mass for a fixed value of the their charge state. But as we increase the
charge state of the negative (positive) ion then the amplitude of the positive
shock profile increases (decreases) when their mass are constant. The effects
of the temperature of electron and positive ion (via $\alpha_{2}$) can be seen
in Fig. 8, and the amplitude of the shock profile enhances (diminishes) with
electron (positive ion) temperature when other plasma parameters are
invariant.

##  5 Conclusion

In our present investigation, we have considered a multi-component magnetized
PIP having static dust grains, non-thermal electrons and positrons. The
Burgers’ equation has been derived by employing reductive perturbation method
[45] for studying DIASHWs. The results that we have found from this
investigation can be summarized as follows:

  * 1.

The negative (positive) shock profile can be exist for the value of
$\alpha_{4}<\alpha_{4c}$ ($\alpha_{4}>\alpha_{4c}$).

  * 2.

The steepness of both positive and negative shock profiles declines with the
increase of $\eta$ without affecting the height.

  * 3.

The increase in oblique angle rises the height of the positive shock profile.

  * 4.

The height of the positive shock wave increases with the number density of
positron.

  * 5.

The temperature of the electrons enhances the amplitude of the shock profile.

The results are applicable in understanding the criteria for the formation of
DIASHWs in astrophysical plasmas, viz., cometary comae [2], upper regions of
Titan’s atmosphere [3, 4, 5], plasmas in the D and F-regions of Earth’s
ionosphere [4, 5, 6], and also in laboratory environments, viz., ($K^{+}$,
$SF_{6}^{-}$) plasma [7, 8], ($Ar^{+}$, $F^{-}$) plasma [9], plasma processing
reactors [10], plasma etching [11], combustion products [11], ($Xe^{+}$,
$F^{-}$) plasma [12], neutral beam sources [13], ($Ar^{+}$, $SF_{6}^{-}$)
plasma [14, 15, 16],($Ar^{+}$, $O_{2}^{-}$) plasma, and Fullerene
($C_{60}^{+}$, $C_{60}^{-}$) plasma [17, 18], etc.

## References

  * [1] P.K. Shukla and A.A. Mamun, Introdustion to Dusty Plasma Physics (Institute of Physics, Bristol, 2002).
  * [2] P.H. Chaizy, et al., Nature (London), 349, 393 (1991).
  * [3] A. J. Coates, et al., Geophys. Res. Lett. 34, L22103 (2007).
  * [4] H. Massey, Negative Ions, 3rd ed., (Cambridge University Press, Cambridge, 1976)
  * [5] R. Sabry, et al., Phys. Plasmas 16, 032302 (2009).
  * [6] H.G. Abdelwahed, et al., Phys. Plasmas 23, 022102 (2016).
  * [7] B. Song, et al., Phys. Fluids B 3, 284 (1991).
  * [8] N. Sato, Plasma Sources Sci. Technol. 3, 395 (1994).
  * [9] Y. Nakamura and I. Tsukabayashi, Phys. Rev. Lett. 52, 2356 (1984).
  * [10] R.A. Gottscho and C.E. Gaebe, IEEE Trans. Plasma Sci. 14, 92 (1986).
  * [11] D.P. Sheehan and N. Rynn, Rev. Sci. lnstrum. 59, 8 (1988).
  * [12] R. Ichiki, et al., Phys. Plasmas 9, 4481 (2002).
  * [13] M. Bacal and G.W. Hamilton, Phys. Rev. Lett. 42, 1538 (1979).
  * [14] A.Y. Wong, D.L. Mamas, and D. Arnush, Phys. Fluids 18, 1489 (1975).
  * [15] J.L. Cooney, et al., Phys. Fluids B 3, 2758 (1991).
  * [16] Y. Nakamura, et al., Plasma Phys. Control. Fusion 39, 105 (1997).
  * [17] W. Oohara and R. Hatakeyama, Phys. Rev. Lett. 91, 205005 (2003).
  * [18] R. Hatakeyama and W. Oohara, Phys. Scripta 116, 101 (2005).
  * [19] P.K. Shukla and V.P. Silin, Phys. Scr. 45, 508 (1992).
  * [20] P.K. Shukla, Phys. Plasmas 7, 1044 (2000).
  * [21] T.K. Baluku, et al., Phys. Plasmas 17, 053702 (2010).
  * [22] S. Sultana, et al., Astrophys. space Sci 351, 581 (2014).
  * [23] S. Yasmin et al., Astrophys Space Sci 343, 245 (2013).
  * [24] A. Rahman, et al., IEEE Transactions on Plasma Sci. 43, 974 (2015).
  * [25] U.M. Abdelsalam, et al., Physics Letters A 372, 4057 (2008).
  * [26] R.A. Carins, et al., et al. Geophys. Res. Lett. 22, 2709 (1995).
  * [27] N.M. Heera, et al., AIP Adv. 11, 055117 (2021); M.H. Rahman,et al., Phys. Plasmas 25, 102118 (2018); J. Akter, et al. Dust-acoustic envelope solitons and rogue waves in an electron depleted plasma. Indian J. Phys (2021). https://doi.org/10.1007/s12648-020-01927-9; N.A. Chowdhury, et al., Phys. plasmas 24, 113701 (2017); M.H. Rahman, et al., Chin. J. Phys. 56, 2061 (2018); N.A. Chowdhury, et al., Plasma Phys. Rep. 45, 459 (2019); S. Jahan, et al., Plasma Phys. Rep. 46, 90 (2020).
  * [28] M.M. Haider, et al., Theoretical Phys. 4, 124 (2019).
  * [29] H.R. Pakzad and K. Javidan, Pranama Journal of Phys. 73, 913 (2009).
  * [30] Y. Ghai, et al., Physics of Plasmas 25, 013704 (2018).
  * [31] A. Sabetkar and D. Dorranian, J. Theor. Appl. Phys. 9, 150 (2015).
  * [32] M. Shahmansouri and A.A. Mamun, J. Plasma Physics 80, 593 (2014).
  * [33] H.K. Malik, et al., J. Taibah Univ. Sci. 14, 417 (2020).
  * [34] C. Bedi, et al., J. Phys.: Conf. Ser. 208, 012037 (2010).
  * [35] N.C. Adhikary, Phys. Lett. A 376, 1460 (2012).
  * [36] B. Sahu, et al., Phys. Plasmas 21, 103701 (2014).
  * [37] A. Atteya, et al., Chin. J. Phys. 56, 1931 (2018).
  * [38] T. Yeashna, et al., Eur. Phys. J. D 75, 135 (2021); B.E. Sharmin, et al., Results Phys. 26, 104373 (2021); S.K. Paul, et al., Pramana J. Phys. 94, 58 (2020); N.A. Chowdhury, et al., Chaos 27, 093105 (2017); N.A. Chowdhury, et al., Contrib. Plasma Phys. 58, 870 (2018); M. Hassan, et al., Commun. Theor. Phys. 71, 1017 (2019); D.M.S. Zaman, et al., High Temp. 58, 789 (2020).
  * [39] S.K. El-Labany, et al., Eur. Phys. J. D 74, 104 (2020).
  * [40] H. Washimi, T. Taniuti, Phys. Rev. Lett. 17, 996 (1966).
  * [41] M.M. Hossen, et al., High Energy Density Phys. 24, 9 (2017).
  * [42] V.I. Karpman, Nonlinear Waves in Dispersive Media, (Pergamon Press, Oxford, 1975).
  * [43] A. Hasegawa, Plasma Instabilities and Nonlinear Effects, (Springer-Verlag, Berlin, 1975).
  * [44] H. Alinejad, Astrophys Space Sci 327, 131 (2010).
  * [45] S. Jahan, et al., Commun. Theor. Phys. 71, 327 (2019); T.I. Rajib, et al., Phys. plasmas 26, 123701 (2019); N. Ahmed, et al., Chaos 28, 123107 (2018); N.A. Chowdhury, et al., Vacuum 147, 31 (2018); R.K. Shikha, et al., Eur. Phys. J. D 73, 177 (2019); S. Banik, et al., Eur. Phys. J. D 75, 43 (2021).