* [59] J Kaas  “Evolution of columns, modules, and domains in the neocortex of primates”  In _Proc Nat’l Acad Sci, USA_ 109, 2012, pp. 10655–10660  DOI: 10.1073/pnas.1201892109
  * [60] J H Kaas  “The evolution of the complex sensory and motor systems of the human brain”  In _Brain Res Bull_ 75.2-4, 2008, pp. 384–390  DOI: 10.1016/j.brainresbull.2007.10.009
  * [61] M Kaschube, M Schnabel and F Wolf  “Self-organization and the selection of pinwheel density in visual cortical development”  In _New J Phys_ 10.1, 2008, pp. 015009
  * [62] M Kaschube, F Wolf, T Geisel and S Löwel  “Genetic influence on quantitative features of neocortical architecture”  In _J Neurosci_ 22.16, 2002, pp. 7206–7217
  * [63] M Kaschube, M Schnabel, F Wolf and S Löwel  “Interareal coordination of columnar architectures during visual cortical development”  In _Proc Natl Acad Sci, USA_ 40.106, 2009, pp. 17205–17210
  * [64] M Kaschube et al.  “The pattern of ocular dominance columns in cat primary visual cortex: intra- and interindividual variability of column spacing and its dependence on genetic background”  In _Eur J Neurosci_ 18, 2003, pp. 3251–3266
  * [65] Matthias Kaschube  “Neural maps versus salt-and-pepper organization in visual cortex”  In _Curr Opin Neurobiol_ 24 Elsevier Ltd, 2014, pp. 95–102  DOI: 10.1016/j.conb.2013.08.017
  * [66] Matthias Kaschube et al.  “Universality in the evolution of orientation columns in the visual cortex”  In _Science_ 330.6007, 2010, pp. 1113–6  DOI: 10.1126/science.1194869
  * [67] W. Keil et al.  “Response to comment on ”Universality in the evolution of orientation columns in the visual cortex””  In _Science_ 336.6080, 2012, pp. 413–413  DOI: 10.1126/science.1206416
  * [68] Wolfgang Keil and Fred Wolf  “Coverage, continuity, and visual cortical architecture”  In _Neural Syst & Circ_ 1.1 BioMed Central Ltd, 2011, pp. 17  DOI: 10.1186/2042-1001-1-17
  * [69] J Klekamp, A A Riedel, C Harper and H J Kretschmann  “Quantitative changes during the postnatal maturation of the human visual cortex”  In _J Neurol Sci_ 103.2, 1991, pp. 136–43
  * [70] R Kretz, G Rager and T T Norton  “Laminar organization of ON and OFF regions and ocular dominance in the striate cortex of the tree shrew (Tupaia belangeri)”  In _J Comp Neurol_ 251.1, 1986, pp. 135–145
  * [71] J O Kriegs et al.  “Evolutionary history of 7SL RNA-derived SINEs in Supraprimates”  In _Trends Genet_ 4.23, 2007, pp. 158–161
  * [72] Jan Ole Kriegs et al.  “Retroposed elements as archives for the evolutionary history of placental mammals”  In _PLoS Biol_ 4.4, 2006, pp. e91  DOI: 10.1371/journal.pbio.0040091
  * [73] G Leuba and R Kraftsik  “Changes in volume, surface estimate, 3-dimensional shape and total number of neurons of the human primary visual cortex from midgestation until old-age”  In _Anat Embryol_ 190.4, 1994, pp. 351–366
  * [74] Simon Levay and Charles D Gilbert  “Laminar patterns of geniculocortical projections in the cat”  In _Brain Res_ 113, 1976, pp. 1–19
  * [75] W. R. Levick and L. N. Thibos  “Orientation bias of cat retinal ganglion cells”  In _Nature_ 286, 1980, pp. 389–390
  * [76] C Malsburg  “Self-organization of orientation sensitive cells in the striate cortex”  In _Kybernetik_ 14.2, 1973, pp. 85–100  URL: http://www.ncbi.nlm.nih.gov/pubmed/7621943
  * [77] James H Marshel, Alfred P Kaye, Ian Nauhaus and Edward M Callaway  “Anterior-posterior direction opponency in the superficial mouse lateral geniculate nucleus.”  In _Neuron_ 76.4 Elsevier Inc., 2012, pp. 713–720  DOI: 10.1016/j.neuron.2012.09.021
  * [78] L M Martinez et al.  “Statistical wiring of thalamic receptive fields optimizes spatial sampling of the retinal image.”  In _Neuron_ 4.81, 2014, pp. 943–956
  * [79] J A Mazer et al.  “Spatial frequency and orientation tuning dynamics in area V1”  In _Proc Nat’l Acad Sci, USA_ 99.3, 2002, pp. 1645–1650
  * [80] R W Meredith et al.  “Impacts of the Cretaceous Terrestrial Revolution and KPg extinction on mammal diversification”  In _Science_ 334.6055, 2011, pp. 521–24
  * [81] K D Miller  “A model for the development of simple cell receptive fields and the ordered arrangement of orientation columns through activity-dependent competition between ON- and OFF-center inputs”  In _J Neurosci_ 14.1, 1994, pp. 409–441
  * [82] J A Movshon, I D Thompson and D J Tolhurst  “Spatial and temporal contrast sensitivity of neurones in areas 17 and 18 of the cat’s visual cortex.”  In _J Physiol_ 283, 1978, pp. 101–120
  * [83] W J Murphy et al.  “Resolution of the early placental mammal radiation using Bayesian phylogenetics”  In _Science_ 294.5550, 2001, pp. 2348–2351
  * [84] I Nauhaus, A Benucci, M Carandini and D L Ringach  “Neuronal selectivity and local map structure in visual cortex”  In _Neuron_ 57, 2008, pp. 673–679
  * [85] Cristopher M Niell and Michael P Stryker  “Highly selective receptive fields in mouse visual cortex”  In _J Neurosci_ 28.30, 2008, pp. 7520–7536  DOI: 10.1523/JNEUROSCI.0623-08.2008
  * [86] Y Nishijima and G Oster  “Moire patterns: Their application to refractive index and refractive index gradient measurements”  In _J Opt Soc Am_ 54, 1964, pp. 1–5
  * [87] K Obermayer and G G Blasdel  “Geometry of orientation and ocular dominance columns in monkey striate cortex”  In _J Neurosci_ 13.10, 1993, pp. 4114–4129
  * [88] K Ohki et al.  “Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex”  In _Nature_ 433, 2005, pp. 597–603
  * [89] K Ohki et al.  “Highly ordered arrangement of single neurons in orientation pinwheels”  In _Nature_ 442, 2006, pp. 925–928
  * [90] M A O’Leary et al.  “Response to comment on ”The placental mammal ancestor and the post-K-Pg radiation of placentals””  In _Science_ 341.6146, 2013, pp. 613
  * [91] M A O’Leary et al.  “The placental mammal ancestor and the post-K-Pg radiation of placentals”  In _Science_ 339.6120, 2013, pp. 662–67
  * [92] S-B Paik  “Developmental models of functional maps in cortex”  In _Biomed Eng Lett_ 4.3, 2014, pp. 223–225
  * [93] S B Paik and D L Ringach  “Link between orientation and retinotopic maps in primary visual cortex”  In _Proc Nat’l Acad Sci, USA_ 109, 2012, pp. 7091–7096
  * [94] Se-Bum Paik and Dario L Ringach  “Retinal origin of orientation maps in visual cortex”  In _Nat Neurosci_ 14.7, 2011, pp. 919–925  DOI: 10.1038/nn.2824
  * [95] A Peters and B R Payne  “Numerical relationships between geniculocortical afferents and pyramidal cell modules in cat primary visual cortex”  In _Cereb Cortex_ 1.3, 1993, pp. 69–78
  * [96] L Reichl et al.  “Coordinated optimization of visual cortical maps (I) Symmetry-based analysis”  In _PLoS Comp Biol_ 8.11, 2012, pp. e1002466
  * [97] L Reichl et al.  “Coordinated optimization of visual cortical maps (II) Numerical studies”  In _PLoS Comp Biol_ 8.11, 2012, pp. e1002756
  * [98] Lars Reichl, Siegrid Löwel and Fred Wolf  “Pinwheel stabilization by ocular dominance segregation”  In _Phys Rev Lett_ 102.20, 2009, pp. 208101  DOI: 10.1103/PhysRevLett.102.208101
  * [99] R C Reid and J M Alonso  “Specificity of monosynaptic connections from thalamus to visual cortex”  In _Nature_ 378.6554 [London: Macmillan Journals], 1869-, 1995, pp. 281–284
  * [100] D L Ringach  “On the origin of the functional architecture of the cortex”  In _PLoS ONE_ 2.2, 2007, pp. e251
  * [101] D L Ringach  “Spatial Structure and Symmetry of Simple-Cell Receptive Fields in Macaque Primary Visual Cortex”  In _J Neurophysiol_ 88.1, 2002, pp. 455–463
  * [102] D L Ringach, R M Shapley and M J Hawken  “Orientation selectivity in macaque V1: diversity and laminar dependence”  In _J Neurosci_ 22.13, 2002, pp. 5639–5651
  * [103] Dario L Ringach  “Haphazard wiring of simple receptive fields and orientation columns in visual cortex.”  In _J Neurophysiol_ 92.1, 2004, pp. 468–476  DOI: 10.1152/jn.01202.2003
  * [104] J D Schall, V H Perry and A G Leventhal  “Retinal ganglion cell dendritic fields in old-world monkeys are oriented radially”  In _Brain Res_ 368.1, 1986, pp. 18–23  URL: http://www.ncbi.nlm.nih.gov/pubmed/3955359
  * [105] M Schnabel, M Kaschube, S Löwel and F Wolf  “Random waves in the brain: Symmetries and defect generation in the visual cortex”  In _Eur Phys J, Special Topics_ 145.1, 2007, pp. 137–157
  * [106] M Schottdorf, S J Eglen, F Wolf and W Keil  “Can retinal ganglion cell dipoles seed iso-orientation domains in the visual cortex?”  In _PLoS ONE_ e86139, 2014
  * [107] L D Selemon and A Begovic  “Stereologic analysis of the lateral geniculate nucleus of the thalamus in normal and schizophrenic subjects”  In _Psychiat Res_ 151.1–2, 2007, pp. 1–10
  * [108] T Shou and A G Leventhal  “Organized arrangement of orientation-sensitive cat’s dorsal lateral geniculate nucleus”  In _J Neurosci_ 9.12, 1989, pp. 4287–4302
  * [109] E L Smith et al.  “Orientation bias of neurons in the lateral geniculate nucleus of macaque monkeys.”  In _Vis Neurosci_ 5.6, 1990, pp. 525–545  URL: http://www.ncbi.nlm.nih.gov/pubmed/2085469
  * [110] R E Soodak  “The retinal ganglion cell mosaic defines orientation columns in striate cortex”  In _Proc Nat’l Acad Sci, USA_ 84.11, 1987, pp. 3936–3940
  * [111] R E Soodak  “Two-dimensional modeling of visual receptive fields using Gaussian subunits”  In _Proc Nat’l Acad Sci, USA_ 83.23, 1986, pp. 9259–9263
  * [112] M S Springer, R W Meredith, E C Teeling and W J Murphy  “Technical comment on ”The placental mammal ancestor and the post-K-Pg radiation of placentals””  In _Science_ 6146.341, 2013, pp. 613
  * [113] N V Swindale  “A model for the formation of orientation columns”  In _Proc Roy Soc B-Biol Sci_ 215.1199, 1982, pp. 211–30  URL: http://www.ncbi.nlm.nih.gov/pubmed/10880004
  * [114] N V Swindale  “Orientation tuning curves: empirical description and estimation of parameters”  In _Biol Cybern_ 78, 1998, pp. 45–56
  * [115] N V Swindale, J A Matsubara and M S Cynader  “Surface organization of orientation and direction selectivity in cat area 18”  In _J Neurosci_ 7.5, 1987, pp. 1414–27
  * [116] K G Thompson, Y Zhou and A G Leventhal  “Direction-sensitive X and Y cells within the A laminae of the cat’s LGNd”  In _Vis Neurosci_ 11.5, 1994, pp. 927–938  URL: http://www.ncbi.nlm.nih.gov/pubmed/7947406
  * [117] W M Usrey, J B Reppas and R C Reid  “Specificity and strength of retinogeniculate connections”  In _J Neurophysiol_ 82.6, 1999, pp. 3527–3540  URL: http://www.ncbi.nlm.nih.gov/pubmed/10601479
  * [118] S D Van Hooser et al.  “Transformation of receptive field properties from lateral geniculate nucleus to superficial V1 in the Tree Shrew”  In _J Neurosci_ 33, 2013, pp. 11494–11505
  * [119] H Wässle, B B Boycott and R B Illing  “Morphology and mosaic of on- and off-beta cells in the cat retina and some functional considerations”  In _Proc Roy Soc B-Biol Sci_ 212.1187, 1981, pp. 177–195
  * [120] H Wässle, Grünert U, Röhrenbeck J and B B Boycott  “Cortical magnification factor and the ganglion cell density of the primate retina”  In _Nature_ 347, 1989, pp. 643–646
  * [121] H Wässle, Grünert U, Röhrenbeck J and B B Boycott  “Retinal ganglion cell density and cortical magnification factor in the primate”  In _Vision Res_ 30.1, 1989, pp. 1897–1911
  * [122] E. W. Weisstein  “Jacobi Theta Functions”  In _MathWorld - A Wolfram Web Resource_ URL: http://mathworld.wolfram.com/JacobiThetaFunctions.html
  * [123] F Wolf  “Symmetry, multistability, and long-range interactions in brain development”  In _Phys Rev Lett_ 95.20, 2005, pp. 208701
  * [124] F Wolf and T Geisel  “Spontaneous pinwheel annihilation during visual development”  In _Nature_ 395, 1998, pp. 73–78
  * [125] F Wolf and T Geisel  “Universality in visual cortical pattern formation”  In _J Physiol_ 97.2-3, 2003, pp. 253–264  DOI: 10.1016/j.jphysparis.2003.09.018
  * [126] F Wörgötter and U T Eysel  “Quantitative determination of orientational and directional components in the response of visual cortical cells to moving stimuli”  In _Biol Cybern_ 355, 1987, pp. 349–355


# On the Cauchy problem for the Hunter-Saxton equation on the line

Weikui $\mbox{Ye}^{1}$ 111email<EMAIL_ADDRESS>and Zhaoyang
$\mbox{Yin}^{1,2}$ 222email<EMAIL_ADDRESS>
${}^{1}\mbox{Department}$ of Mathematics, Sun Yat-sen University,
Guangzhou, 510275, China
${}^{2}\mbox{Faculty}$ of Information Technology,
Macau University of Science and Technology, Macau, China

###### Abstract

In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS)
equation on the line. Firstly, we establish the local well-posedness for the
integral form of the (HS) equation by constructing some special spaces
$E^{s}_{p,r}$, which mix Lebesgue spaces and homogeneous Besov spaces. Then we
present a global existence result and provide a sufficient condition for
strong solutions to blow up in finite time for the equation. Finally, we give
the ill-posedness and the unique continuation of the Hunter-Saxton equation.

Keywords: Hunter-Saxton equation, Local well-posedness, Global existence, Blow
up, Ill-posedness, Unique continuation.
Mathematics Subject Classification: 35G25, 35A01, 35L03, 35L05, 35L60

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Local well-posedness
  4. 4 Blow-up and global existence
  5. 5 ill-posedness
  6. 6 Unique continuation

##  1 Introduction

In this paper, we consider the following Hunter-Saxton $(HS)$ equation:

$\displaystyle(u_{t}+uu_{x})_{x}=\frac{1}{2}u_{x}^{2}.$ (1.1)

The equation (1.1) was derived by Hunter and Saxton as an asymptotic model of
liquid crystals [16, 20]. The $HS$ equation is completely integrable [20, 17]
and has a bi-Hamiltonian structure [19]. Local well-posedness and blow-up
phenomena for the Cauchy problems of the (HS) equation on the circle were
studied in [16, 24]. Global weak solutions of the (HS) equation was
investigated in [4, 5].

In particular, the (HS) equation is the limit form of the well-known Camassa-
Holm (CH) equation:

$(1-\partial_{x}^{2})u_{t}=3uu_{x}-2u_{x}u_{xx}-uu_{xxx}.$

Local well-posedness and ill-posedness for the Cauchy problem of the (CH)
equation were investigated in [9, 22, 21]. Blow-up phenomena and global
existence of strong solutions were discussed in [8, 6, 9, 7]. The existence of
global weak solutions and dissipative solutions were also investigated in [23,
3, 2]. Moreover, the existence and uniqueness of the global conservative
solutions on the line were studied in [10, 11]. While the existence and
uniqueness of the global conservative solutions on the circle were studied in
[14, 13].

However, the Cauchy problem of the Hunter-Saxton on the line has not been
studied yet. In this paper we consider the Cauchy problem for the Hunter-
Saxton (HS) on the line:

$\left\\{\begin{array}[]{l}u_{xt}+(uu_{x})_{x}=\frac{1}{2}u_{x}^{2},\\\
u(0,x)=u_{0}(x),\quad x\in\mathbb{R}.\end{array}\right.$ (1.2)

Taking $\int_{\infty}^{x}dz$ to the Hunter-Saxton equation (1.2) and letting
$g(t)$ is a bounded function, we get

$\left\\{\begin{array}[]{l}u_{t}+uu_{x}=\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz+g(t),\\\
u(0,x)=u_{0}(x),\quad x\in\mathbb{R}.\end{array}\right.$ (1.3)

In fact, the mainly difficulty is that the term
$\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz$ is not bounded in the
inhomogeneous Besov spaces $B^{s}_{p,r}$, even the Lebesgue spaces $L^{p}$
$(1\leq p<\infty)$. Since $u^{2}_{x}\geq 0$ and
$\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz$ is monotonic increasing, one
can’t consider (1.3) in $B^{s}_{p,r}$ on the line. To overcome this
difficulty, we would like to study (1.3) in some new spaces $E^{s}_{p,r}$
which mix Lebesgue spaces and homogeneous Besov spaces. In this way, one will
get a bounded iterative sequence in $E^{s}_{p,r}$ and finally obtain a
solution for the Cauchy problem of (1.3).

The remaining part of the paper is organized as follows. In Section 2, we
introduce some useful preliminaries. In Section 3, we prove the local well-
posedness of (1.3) in some special space $E^{s}_{p,r}$, the main approach is
based on the Littlewood-Paley theory and transport equations theory. In
Section 4, we obtain a global existence result and give a sufficient condition
for strong solutions to blow up in finite time. In Section 5, we give the ill-
posedness of (1.3) in the special Besov space
$L^{\infty}\cap\dot{B}^{1}_{p,r}\cap\dot{B}^{1+\frac{d}{p}}_{p,r},r>1$. In
Section 6, we give the unique continuation of (1.3) when
$g(t):=C\int_{-\infty}^{x}u_{x}^{2}dz$.

##  2 Preliminaries

In this section, we will recall some propositons on the Littlewood-Paley
decomposition and Besov spaces.

######  Proposition 2.1.

[1] Let $\mathcal{C}$ be the annulus
$\\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$. There exist
radial functions $\chi$ and $\varphi$, valued in the interval $[0,1]$,
belonging respectively to $\mathcal{D}(B(0,\frac{4}{3}))$ and
$\mathcal{D}(\mathcal{C})$, and such that

$\forall\xi\in\mathbb{R}^{d},\ \chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1,$
$\forall\xi\in\mathbb{R}^{d}\backslash\\{0\\},\
\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,$ $|j-j^{\prime}|\geq
2\Rightarrow\mathrm{Supp}\ \varphi(2^{-j}\cdot)\cap\mathrm{Supp}\
\varphi(2^{-j^{\prime}}\cdot)=\emptyset,$ $j\geq 1\Rightarrow\mathrm{Supp}\
\chi(\cdot)\cap\mathrm{Supp}\ \varphi(2^{-j}\cdot)=\emptyset.$

The set $\widetilde{\mathcal{C}}=B(0,\frac{2}{3})+\mathcal{C}$ is an annulus,
and we have

$|j-j^{\prime}|\geq 5\Rightarrow 2^{j}\mathcal{C}\cap
2^{j^{\prime}}\widetilde{\mathcal{C}}=\emptyset.$

Further, we have

$\forall\xi\in\mathbb{R}^{d},\ \frac{1}{2}\leq\chi^{2}(\xi)+\sum_{j\geq
0}\varphi^{2}(2^{-j}\xi)\leq 1,$
$\forall\xi\in\mathbb{R}^{d}\backslash\\{0\\},\
\frac{1}{2}\leq\sum_{j\in\mathbb{Z}}\varphi^{2}(2^{-j}\xi)\leq 1.$

Denote $\mathcal{F}$ by the Fourier transform and $\mathcal{F}^{-1}$ by its
inverse. Let $u$ be a tempered distribution in
$\mathcal{S}^{\prime}_{h}(\mathbb{R}^{d})$. For all $j\in\mathbb{Z}$, define

$\dot{\Delta}_{j}u=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\mathcal{F}u)\,\
\quad\dot{S}_{j}u=\sum_{j^{\prime}<j}\Delta_{j^{\prime}}u.$

Then the Littlewood-Paley decomposition is given as follows:

$u=\sum_{j\in\mathbb{Z}}\Delta_{j}u\quad\text{in}\
\mathcal{S}^{\prime}_{h}(\mathbb{R}^{d}).$

Let $s\in\mathbb{R},\ 1\leq p,r\leq\infty.$ The homogeneous Besov space
$\dot{B}^{s}_{p,r}(\mathbb{R}^{d})$ is defined by

$\dot{B}^{s}_{p,r}=\dot{B}^{s}_{p,r}(\mathbb{R}^{d})=\\{u\in
S^{\prime}_{h}(\mathbb{R}^{d}):\|u\|_{\dot{B}^{s}_{p,r}(\mathbb{R}^{d})}=\Big{\|}(2^{js}\|\dot{\Delta}_{j}u\|_{L^{p}})_{j}\Big{\|}_{l^{r}(\mathbb{Z})}<\infty\\}.$

Let $u$ be a tempered distribution in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$.
For all $j\in\mathbb{Z}$, define

$\Delta_{j}u=0\,\ \text{if}\,\
j\leq-2,\quad\Delta_{-1}u=\mathcal{F}^{-1}(\chi\mathcal{F}u),\quad\Delta_{j}u=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\mathcal{F}u)\,\
\text{if}\,\ j\geq 0,\quad S_{j}u=\sum_{j^{\prime}<j}\Delta_{j^{\prime}}u.$

Then the Littlewood-Paley decomposition is given as follows:

$u=\sum_{j\in\mathbb{Z}}\Delta_{j}u\quad\text{in}\
\mathcal{S}^{\prime}(\mathbb{R}^{d}).$

Let $s\in\mathbb{R},\ 1\leq p,r\leq\infty.$ The nonhomogeneous Besov space
$B^{s}_{p,r}(\mathbb{R}^{d})$ is defined by

$B^{s}_{p,r}=B^{s}_{p,r}(\mathbb{R}^{d})=\\{u\in
S^{\prime}(\mathbb{R}^{d}):\|u\|_{B^{s}_{p,r}(\mathbb{R}^{d})}=\Big{\|}(2^{js}\|\Delta_{j}u\|_{L^{p}})_{j}\Big{\|}_{l^{r}(\mathbb{Z})}<\infty\\}.$

There are some properties about Besov spaces. For the homogeneous Besov space,
we have

######  Proposition 2.2.

[1, 12] Let $s\in\mathbb{R},\ 1\leq p,p_{1},p_{2},r,r_{1},r_{2}\leq\infty$
with $s<\frac{d}{p}ors=\frac{d}{p},r=1$.
(1) $\dot{B}^{s}_{p,r}$ is a Banach space, and is continuously embedded in
$\mathcal{S}^{\prime}_{h}$.
(2) If $r<\infty$, then
$\lim\limits_{j\rightarrow\infty}\|S_{j}u-u\|_{\dot{B}^{s}_{p,r}}=0$.
(3) If $p_{1}\leq p_{2}$ and $r_{1}\leq r_{2}$, then
$\dot{B}^{s}_{p_{1},r_{1}}\hookrightarrow\dot{B}^{s-d(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}.$
(4) If $p\in[1,2]$, then $\dot{B}^{s}_{p,p}\hookrightarrow L^{p}$. If
$p\in[2,\infty)$, then $\dot{B}^{s}_{p,2}\hookrightarrow L^{p}$. If
$p=\infty$, then $\dot{B}^{s}_{\infty,1}\hookrightarrow L^{\infty}\quad.$
(5) Fatou property: if $(u_{n})_{n\in\mathbb{N}}$ is a bounded sequence in
$\dot{B}^{s}_{p,r}$, then an element $u\in\dot{B}^{s}_{p,r}$ and a subsequence
$(u_{n_{k}})_{k\in\mathbb{N}}$ exist such that

$\lim_{k\rightarrow\infty}u_{n_{k}}=u\ \text{in}\
\mathcal{S}^{\prime}_{h}\quad\text{and}\quad\|u\|_{\dot{B}^{s}_{p,r}}\leq
C\liminf_{k\rightarrow\infty}\|u_{n_{k}}\|_{\dot{B}^{s}_{p,r}}.$

(6) Let $m\in\mathbb{R}$ and $f$ be a $S^{m}$-mutiplier, (i.e. f is a smooth
function and satisfies that $\forall\alpha\in\mathbb{N}^{d}$, $\exists
C=C(\alpha)$, such that $|\partial^{\alpha}f(\xi)|\leq C|\xi|^{m-|\alpha|},\
\forall\xi\in\mathbb{R}^{d})$. Then the operator
$f(D)=\mathcal{F}^{-1}(f\mathcal{F})$ is continuous from $B^{s}_{p,r}$ to
$B^{s-m}_{p,r}$.
(7) If $s>0$, then $L^{p}\cap\dot{B}^{s}_{p,r}=B^{s}_{p,r}$

For the nonhomogeneous Besov space, we have

######  Proposition 2.3.

[1, 12] Let $s\in\mathbb{R},\ 1\leq p,p_{1},p_{2},r,r_{1},r_{2}\leq\infty.$
(1) $B^{s}_{p,r}$ is a Banach space, and is continuously embedded in
$\mathcal{S}^{\prime}$.
(2) If $r<\infty$, then
$\lim\limits_{j\rightarrow\infty}\|S_{j}u-u\|_{B^{s}_{p,r}}=0$. If
$p,r<\infty$, then $C_{0}^{\infty}$ is dense in $B^{s}_{p,r}$.
(3) If $p_{1}\leq p_{2}$ and $r_{1}\leq r_{2}$, then
$B^{s}_{p_{1},r_{1}}\hookrightarrow
B^{s-d(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}.$ If $s_{1}<s_{2}$,
then the embedding $B^{s_{2}}_{p,r_{2}}\hookrightarrow B^{s_{1}}_{p,r_{1}}$ is
locally compact.
(4) If $p\in[1,2]$, then $B^{s}_{p,p}\hookrightarrow L^{p}$. If
$p\in[2,\infty)$, then $B^{s}_{p,2}\hookrightarrow L^{p}$. If $p=\infty$, then
$B^{s}_{\infty,1}\hookrightarrow L^{\infty}\quad.$
(5) Fatou property: if $(u_{n})_{n\in\mathbb{N}}$ is a bounded sequence in
$B^{s}_{p,r}$, then an element $u\in B^{s}_{p,r}$ and a subsequence
$(u_{n_{k}})_{k\in\mathbb{N}}$ exist such that

$\lim_{k\rightarrow\infty}u_{n_{k}}=u\ \text{in}\
\mathcal{S}^{\prime}\quad\text{and}\quad\|u\|_{B^{s}_{p,r}}\leq
C\liminf_{k\rightarrow\infty}\|u_{n_{k}}\|_{B^{s}_{p,r}}.$

(6) Let $m\in\mathbb{R}$ and $f$ be a $S^{m}$-mutiplier, (i.e. f is a smooth
function and satisfies that $\forall\alpha\in\mathbb{N}^{d}$, $\exists
C=C(\alpha)$, such that $|\partial^{\alpha}f(\xi)|\leq
C(1+|\xi|)^{m-|\alpha|},\ \forall\xi\in\mathbb{R}^{d})$. Then the operator
$f(D)=\mathcal{F}^{-1}(f\mathcal{F})$ is continuous from $B^{s}_{p,r}$ to
$B^{s-m}_{p,r}$.

We then introduce some useful lammas in nonhomogeneous Besov space(homogeneous
Besov space is similar).

######  Lemma 2.4.

[1, 12] (1) If $s_{1}<s_{2}$, $\theta\in(0,1)$, and $(p,r)$ is in
$[1,\infty]^{2}$, then we have

$\|u\|_{B^{\theta
s_{1}+(1-\theta)s_{2}}_{p,r}}\leq\|u\|_{B^{s_{1}}_{p,r}}^{\theta}\|u\|_{B^{s_{2}}_{p,r}}^{1-\theta}.$

(2) If $s\in\mathbb{R},\ 1\leq p\leq\infty,\ \varepsilon>0$, a constant
$C=C(\varepsilon)$ exists such that

$\|u\|_{B^{s}_{p,1}}\leq
C\|u\|_{B^{s}_{p,\infty}}\ln\Big{(}e+\frac{\|u\|_{B^{s+\varepsilon}_{p,\infty}}}{\|u\|_{B^{s}_{p,\infty}}}\Big{)}.$

######  Lemma 2.5.

[1, 12] (1) For any $s>0$ and any $(p,r)$ in $[1,\infty]^{2}$, the space
$L^{\infty}\cap B^{s}_{p,r}$ is an algebra, and a constant $C=C(s,d)$ exists
such that

$\|uv\|_{B^{s}_{p,r}}\leq
C(\|u\|_{L^{\infty}}\|v\|_{B^{s}_{p,r}}+\|u\|_{B^{s}_{p,r}}\|v\|_{L^{\infty}}).$

(2) If $1\leq p,r\leq\infty,\ s_{1}\leq s_{2},\
s_{2}>\frac{d}{p}(s_{2}\geq\frac{d}{p}\ \text{if}\ r=1)$ and
$s_{1}+s_{2}>\max(0,\frac{2d}{p}-d)$, there exists $C=C(s_{1},s_{2},p,r,d)$
such that

$\|uv\|_{B^{s_{1}}_{p,r}}\leq
C\|u\|_{B^{s_{1}}_{p,r}}\|v\|_{B^{s_{2}}_{p,r}}.$

(3) If $1\leq p\leq 2$, there exists $C=C(p,d)$ such that

$\|uv\|_{B^{\frac{d}{p}-d}_{p,\infty}}\leq
C\|u\|_{B^{\frac{d}{p}-d}_{p,\infty}}\|v\|_{B^{\frac{d}{p}}_{p,1}}.$

Now we state some useful results in the transport equation theory, which are
crucial to the proofs of our main theorem later.

$\left\\{\begin{array}[]{l}f_{t}+v\cdot\nabla f=g,\ x\in\mathbb{R}^{d},\
t>0,\\\ f(0,x)=f_{0}(x).\end{array}\right.$ (2.1)

######  Lemma 2.6.

[1, 18] Let $s\in\mathbb{R},\ 1\leq p,r\leq\infty$. There exists a constant
$C$ such that for all solutions $f\in L^{\infty}([0,T];B^{s}_{p,r})$ of (2.1)
in one dimension with initial data $f_{0}$ in $B^{s}_{p,r}$, and $g$ in
$L^{1}([0,T];B^{s}_{p,r})$, we have, for a.e. $t\in[0,T]$,

$\|f(t)\|_{B^{s}_{p,r}}\leq
e^{CV(t)}\Big{(}\|f_{0}\|_{B^{s}_{p,r}}+\int_{0}^{t}e^{-CV(t^{\prime})}\|g(t^{\prime})\|_{B^{s}_{p,r}}dt^{\prime}\Big{)}$

with

$V^{\prime}(t)=\left\\{\begin{array}[]{ll}\|\nabla
v\|_{B^{s+1}_{p,r}},&\text{if}\ s>\max(-\frac{1}{2},\frac{1}{p}-1),\\\
\|\nabla v\|_{B^{s}_{p,r}},&\text{if}\ s>\frac{1}{p}\ \text{or}\
(s=\frac{1}{p},\ p<\infty,\ r=1),\end{array}\right.$

and when $s=\frac{1}{p}-1,\ 1\leq p\leq 2,\ r=\infty,\ \text{and}\
V^{\prime}(t)=\|\nabla v\|_{B^{\frac{1}{p}}_{p,1}}$.

######  Lemma 2.7.

[1] Let $1\leq p\leq p_{1}\leq\infty,\ 1\leq r\leq\infty,\
s>-d\min(\frac{1}{p_{1}},\frac{1}{p^{\prime}})$. Let $f_{0}\in B^{s}_{p,r}$,
$g\in L^{1}([0,T];B^{s}_{p,r})$, and let $v$ be a time-dependent vector field
such that $v\in L^{\rho}([0,T];B^{-M}_{\infty,\infty})$ for some $\rho>1$ and
$M>0$, and

$\begin{array}[]{ll}\nabla v\in
L^{1}([0,T];B^{\frac{d}{p_{1}}}_{p_{1},\infty}),&\ \text{if}\
s<1+\frac{d}{p_{1}},\\\ \nabla v\in L^{1}([0,T];B^{s-1}_{p,r}),&\ \text{if}\
s>1+\frac{d}{p_{1}}\ or\ (s=1+\frac{d}{p_{1}}\ and\ r=1).\end{array}$

Then the equation (2.1) has a unique solution $f$ in
-the space $C([0,T];B^{s}_{p,r})$, if $r<\infty$,
-the space $\Big{(}\bigcap_{s^{\prime}<s}C([0,T];B^{s^{\prime}}_{p,\infty})\Big{)}\bigcap C_{w}([0,T];B^{s}_{p,\infty})$, if $r=\infty$.

######  Lemma 2.8.

[18] Let $1\leq p\leq\infty,\ 1\leq r<\infty,\ s>\frac{d}{p}\ (or\
s=\frac{d}{p},\ p<\infty,\ r=1)$. Denote
$\bar{\mathbb{N}}=\mathbb{N}\cup\\{\infty\\}$. Let
$(v^{n})_{n\in\bar{\mathbb{N}}}\in C([0,T];B^{s+1}_{p,r})$. Assume that
$(f^{n})_{n\in\bar{\mathbb{N}}}$ in $C([0,T];B^{s}_{p,r})$ is the solution to

$\left\\{\begin{array}[]{l}f^{n}_{t}+v^{n}\cdot\nabla f^{n}=g,\
x\in\mathbb{R}^{d},\ t>0,\\\ f^{n}(0,x)=f_{0}(x)\end{array}\right.$ (2.2)

with initial data $f_{0}\in B^{s}_{p,r},\ g\in L^{1}([0,T];B^{s}_{p,r})$ and
that for some $\alpha\in L^{1}([0,T])$,
$\sup\limits_{n\in\bar{\mathbb{N}}}\|v^{n}(t)\|_{B^{s+1}_{p,r}}\leq\alpha(t)$.
If $v^{n}\rightarrow v^{\infty}$ in $L^{1}([0,T];B^{s}_{p,r})$, then
$f^{n}\rightarrow f^{\infty}$ in $C([0,T];B^{s}_{p,r})$.

##  3 Local well-posedness

In this section, we establish local well-posedness of (1.3) in Besov spaces.
In fact, though $u\in B^{s}_{p,r}$ $(1\leq p<\infty,s>0)$, and u is not a
constant, the right hand side of (1.3)

$\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz+g(t)\notin B^{s}_{p,r}.$

For example, if $u\in H^{2}$, let
$g(t)=0,f(t,x):=\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz$, we have

$f_{x}(t,x)=\frac{1}{2}u_{x}^{2}(z)\geq 0.$

This implies $f(t,x)\notin L^{p}(\mathbb{R})$ $(1\leq p<\infty)$, let along
$f(t,x)\in B^{s}_{p,r}$. To overcome this difficulty, we firstly introduce the
following new function spaces.

######  Definition 3.1.

Let $s\geq 2$ and $1\leq p,r\leq\infty.$ Set

$E^{s}_{p,r}\triangleq\left\\{\begin{array}[]{l}L^{\infty}\cap\dot{B}^{s-1}_{p,r}\cap\dot{B}^{s-2}_{p,r}\cap\dot{W}^{1,q},\quad\quad\quad
when\quad s>2,\\\
E^{2}_{p,r},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
when\quad s=2,\end{array}\right.$ (3.1)
$\bar{E}^{s}_{p,r}\triangleq\left\\{\begin{array}[]{l}L^{\infty}\cap\dot{B}^{s-1}_{p,r},\quad\quad\quad\quad\quad
when\quad s>2,\\\ \bar{E}^{2}_{p,r},\quad\quad\quad\quad\quad\quad\quad\quad
when\quad s=2,\end{array}\right.$ (3.2)

where

$E^{2}_{p,r}\triangleq\left\\{\begin{array}[]{l}L^{\infty}\cap\dot{B}^{1}_{p,r}\cap\dot{B}^{2}_{p,r}\cap\dot{W}^{1,q},\quad
r=p,\quad when\quad 1\leq p\leq 2,\\\
L^{\infty}\cap\dot{B}^{1}_{p,r}\cap\dot{B}^{2}_{p,r}\cap\dot{W}^{1,q},\quad
r=2,\quad when\quad 2\leq p\leq\infty,\end{array}\right.$ (3.3)
$\bar{E}^{2}_{p,r}\triangleq\left\\{\begin{array}[]{l}L^{\infty}\cap\dot{B}^{1}_{p,r},\quad
r=p,\quad when\quad 1\leq p\leq 2,\\\ L^{\infty}\cap\dot{B}^{1}_{p,r},\quad
r=2,\quad when\quad 2\leq p\leq\infty,\end{array}\right.$ (3.4)

and $\frac{1}{p}+\frac{1}{q}=1$.

For example, if $s=p=r=2$, then
$E^{2}_{2,2}=L^{\infty}\cap\dot{H}^{1}\cap\dot{H}^{2}$, which implies $u\in
B^{\frac{3}{2}}_{\infty,2}$ and $u_{x}\in H^{1}$ by proposition 2.2
($\dot{H}^{2}\hookrightarrow\dot{B}^{\frac{3}{2}}_{\infty,2},L^{\infty}\cap\dot{B}^{\frac{3}{2}}_{\infty,2}=B^{\frac{3}{2}}_{\infty,2}$).
If $s=2,p=r=1$,
$E^{2}_{1,1}=L^{\infty}\cap\dot{B}^{1}_{1,1}\cap\dot{B}^{2}_{1,1}$, which
implies $u\in B^{1}_{\infty,1}$ and $u_{x}\in B^{1}_{1,1}$. Actually, when
$1\leq p\leq 2$, we have
$\dot{B}^{s-1}_{p,p}\cap\dot{B}^{s-2}_{p,p}\subset\dot{W}^{1,q}$, so the space
$\dot{W}^{1,q}$ is unnecessary. Moreover, we can easily prove that both
$E^{2}_{p,r}$ and $\bar{E}^{s}_{p,r}$ are Bananch spaces.

Here is our main result.

######  Theorem 3.2.

Let $u_{0}\in E^{s}_{p,r}$. Then there exists a time $T>0$ such that (1.3) has
a unique solution u in $C([0,T];E^{s}_{p,r})\cap
C^{1}([0,T];\bar{E}^{s}_{p,r}).$ Moreover the solution depends continuously on
the initial data.

###### Proof.

We use six steps to prove Theorem 3.2. Without loss of generality, we consider
the case $g(t)=0$ in (1.3) for $g(t)$ is just a bound term independent of x.
And for simplicity, we first consider the case
$E^{2}_{2,2}=L^{\infty}\cap\dot{H}^{1}\cap\dot{H}^{2}=B^{\frac{3}{2}}_{\infty,2}\cap\dot{H}^{1}\cap\dot{H}^{2}.$

Step one. Constructing approximate solution.

We firstly set $u^{0}\triangleq 0,$ and define a sequence
$(u^{n})_{n\in\mathbb{N}}$ of smooth functions by solving the following linear
transport equations:

$\left\\{\begin{array}[]{l}u_{t}^{n+1}+u^{n}u_{x}^{n+1}=\int_{-\infty}^{x}\frac{1}{2}(u_{x}^{n})^{2}(z)dz,\\\
u^{n+1}|_{t=0}=u_{0}.\end{array}\right.$ (3.5)

Define

$G^{n}=\int_{-\infty}^{x}\frac{1}{2}(u_{x}^{n})^{2}(z)dz.$

We assume that $u^{n}\in C([0,T];E^{2}_{2,2})\cap C([0,T];\bar{E}^{2}_{2,2})$
for all $T>0$, then we have

$\|G^{n}\|_{L^{\infty}}\leq C\|u^{n}_{x}\|^{2}_{L^{2}}\leq
C\|u^{n}\|^{2}_{E^{2}_{2,2}},$ $\|G^{n}_{x}\|_{L^{2}}\leq
C\|u^{n}_{x}\|_{L^{\infty}}\|u^{n}_{x}\|_{L^{2}}\leq
C\|u^{n}\|^{2}_{E^{2}_{2,2}},$ $\|G^{n}_{xx}\|_{L}^{2}\leq
C\|u^{n}_{x}\|_{L^{\infty}}\|u^{n}_{xx}\|_{L^{2}}\leq
C\|u^{n}\|^{2}_{E^{2}_{2,2}}.$

This implies that

$\displaystyle\|G^{n}\|_{E^{2}_{2,2}}\leq C\|u^{n}\|^{2}_{E^{2}_{2,2}}.$ (3.6)

So $G^{n}\in L^{\infty}([0,T];E^{2}_{2,2})$. Similar to the proof of Lemma
2.7, we deduce that (3.5) has a global solution $u^{n+1}$ which belongs to
$C([0,T];E^{2}_{2,2})\cap C([0,T];\bar{E}^{2}_{2,2})$ for all $T>0$.

Step two. Uniform bounds.

Using Lemma 2.6, we have

$\|u^{n+1}(t)\|_{L^{\infty}}\leq
e^{C\int_{0}^{t}\|u^{n}\|_{E^{2}_{2,2}}dt^{\prime}}\Big{(}\|u_{0}\|_{E^{2}_{2,2}}+\int_{0}^{t}e^{-C\int_{0}^{t^{\prime}}\|u^{n}\|_{E^{2}_{2,2}}dt^{\prime\prime}}\|G^{n}\|_{E^{2}_{2,2}}dt^{\prime}\Big{)},$
(3.7) $\|u^{n+1}_{x}(t)\|_{H^{1}}\leq
e^{C\int_{0}^{t}\|u^{n}\|_{E^{2}_{2,2}}dt^{\prime}}\Big{(}\|u_{0}\|_{E^{2}_{2,2}}+\int_{0}^{t}e^{-C\int_{0}^{t^{\prime}}\|u^{n}\|_{E^{2}_{2,2}}dt^{\prime\prime}}\|G^{n}\|_{E^{2}_{2,2}}dt^{\prime}\Big{)},$
(3.8)

Combining (3.7), (3.8) and (3.6), we get

$\displaystyle\|u^{n+1}(t)\|_{E^{2}_{2,2}}\leq
e^{C\int_{0}^{t}\|u^{n}\|_{E^{2}_{2,2}}dt^{\prime}}\Big{(}\|u_{0}\|_{E^{2}_{2,2}}+\int_{0}^{t}e^{-C\int_{0}^{t^{\prime}}\|u^{n}\|_{E^{2}_{2,2}}dt^{\prime\prime}}\|u^{n}\|^{2}_{E^{2}_{2,2}}dt^{\prime}\Big{)}$
. (3.9)

Then, we fix a $T>0$ such that $2C^{2}T\|u_{0}\|_{E^{2}_{2,2}}<1$ and suppose
that

$\forall t\in[0,T],\
\|u^{n}(t)\|_{E^{2}_{2,2}}\leq\frac{C\|m_{0}\|_{E^{2}_{2,2}}}{(1-2C^{2}t\|u_{0}\|_{E^{2}_{2,2}})}.$
(3.10)

Plugging (3.10) into (3.9) and using a simple calculation yield

$\displaystyle\|u^{n+1}(t)\|_{E^{2}_{2,2}}$ $\displaystyle\leq
C\|u_{0}\|_{E^{2}_{2,2}}(1-2C^{2}t\|u_{0}\|_{E^{2}_{2,2}})^{-\frac{1}{2}}\Big{(}1+C^{2}\|u_{0}\|_{E^{2}_{2,2}}\int_{0}^{t}(1-2C^{2}t\|u_{0}\|_{E^{2}_{2,2}})^{-\frac{3}{2}}dt^{\prime}\Big{)}$
(3.11)
$\displaystyle\leq\frac{C\|u_{0}\|_{E^{2}_{2,2}}}{(1-2C^{2}t\|u_{0}\|_{E^{2}_{2,2}})}\approx
C_{u_{0}}.$ (3.12)

Therefore, $\\{u^{n}\\}_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}([0,T];{E^{2}_{2,2}})$.

Step three. Cauchy sequence.

We are going to prove that $\\{u^{n}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence
in $L^{\infty}([0,T];L^{\infty})$ and $(u_{x}^{n})_{n\in\mathbb{N}}$ is a
Cauchy sequence in $L^{\infty}([0,T];L^{2})$. For this purpose, set
$u^{n+m+1}-u^{n+1}$, we obtain

$\partial_{t}(u^{n+m+1}-u^{n+1})+u^{n+m+1}\partial_{x}(u^{n+m+1}-u^{n+1})=-(u^{n+m}-u^{n})u^{n+1}_{x}+\int_{\infty}^{x}\frac{1}{2}(u^{n+m}_{x}-u^{n}_{x})(u^{n+m}_{x}+u^{n}_{x})dz.$
(3.13)

By virtue to Lemma 2.6 and (3.11), we have

$\displaystyle\|u^{n+m+1}-u^{n+1}\|_{L^{\infty}}\leq
C_{u_{0}}\int_{0}^{t}\|u^{n+m}-u^{n}\|_{\dot{H}^{1}\cap L^{\infty}}ds.$ (3.14)

and

$\displaystyle\|u^{n+m+1}-u^{n+1}\|_{\dot{H}^{1}}$
$\displaystyle=\|(u^{n+m+1}-u^{n+1})_{x}\|_{L^{2}}$ $\displaystyle\leq
C_{u_{0}}\int_{0}^{t}\|u^{n+m}-u^{n}\|_{\dot{H}^{1}\cap L^{\infty}}ds.$ (3.15)

Combining (3.14) and (3) yields that

$\displaystyle\|u^{n+m+1}-u^{n+1}\|_{\dot{H}^{1}\cap L^{\infty}}$
$\displaystyle\leq C_{u_{0}}\int_{0}^{t}\|u^{n+m}-u^{n}\|_{\dot{H}^{1}\cap
L^{\infty}}ds$
$\displaystyle\leq\frac{(TC_{u_{0}})^{n+1}}{(n+1)!}\|u^{m}\|_{E^{2}_{2,2}}$
$\displaystyle\leq C_{u_{0},T}2^{-n}$ (3.16)

This implies $\\{u^{n}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence in
$C([0,T];L^{\infty}\cap\dot{H}^{1})$. Since $(u^{n})_{n\in\mathbb{N}}$ is
bounded in $L^{\infty}([0,T];E^{2}_{2,2})$. Using interpolation inequalities
implies that

$u^{n}\rightarrow u\quad in\quad
B^{\frac{3}{2}-\varepsilon}_{\infty,2},\quad\varepsilon>0$
$u^{n}_{x}\rightarrow\bar{u}_{x}\quad in\quad
H^{1-\varepsilon},\quad\varepsilon>0$

Step four. Convergence.

We then prove that $u$ in $E^{2}_{2,2}$ and satisfies (1.3). Applying the
Fatou property we deduce that

$\left\\{\begin{array}[]{l}u^{n}\quad is\quad bounded\quad in\quad
B^{\frac{3}{2}}_{\infty,2}\quad\Longrightarrow u_{n}\rightharpoonup u\in
B^{\frac{3}{2}}_{\infty,2},\quad u_{nx}\rightharpoonup u_{x}\in
B^{\frac{1}{2}}_{\infty,2}\\\ u^{n}_{x}\quad is\quad bounded\quad in\quad
H^{1}\quad\Longrightarrow u_{nx}\rightharpoonup\bar{u}_{x}\in H^{1}\subset
B^{\frac{1}{2}}_{\infty,2}.\end{array}\right.$ (3.17)

Because $H^{1}\subset B^{\frac{1}{2}}_{\infty,2}$, we get $\bar{u}_{x}=u_{x}$.
In fact,

$<\bar{u_{x}}-u_{x},\varphi>=<\bar{u_{x}}-u^{n}_{x},u_{x}>+<u_{x}-u^{n}_{x},\varphi>\rightarrow
0,\quad\forall\varphi\in C^{\infty}_{0}.$

Then we may pass to the limit in (3.5) easily and conclude that $u$ is indeed
a solution of (1.3) in the sense of distributions.

Finally, as $u$ belongs to $L^{\infty}([0,T];E^{2}_{2,2})$, the right-hand
side of (1.3) also belongs to $L^{\infty}([0,T];L^{\infty}\cap\dot{H}^{1})$,
which implies $u_{t}$ is in $C([0,T];L^{\infty}\cap\dot{H}^{1})$. Similar to
the proof of Lemma 2.7 in [1], we can easily deduce that $u$ belongs to
$C([0,T];E^{2}_{2,2})\cap C^{1}([0,T];\bar{E}^{2}_{2,2})$.

Step five. Uniqueness and continuous dependence.

We will prove the uniqueness of solutions to (1.3) next. This proof is based
on the way we have in Step 3. Suppose that $(m_{1},m_{2})$ are two solutions
of (1.3), set $w=m_{1}-m_{2}$, we obtain

$\partial_{t}w+m_{1}\partial_{x}w=-wm_{2x}+\int_{-\infty}^{x}\frac{1}{2}w_{x}(m_{1}+m_{2})dz,$

By virtue to Lemma 2.6, we have

$\displaystyle\|w(t)\|_{L^{\infty}}\leq\|w(0)\|_{L^{\infty}}+C_{u_{0}}\int_{0}^{t}\|w\|_{\dot{H}^{1}\cap
L^{\infty}}ds.$ (3.18)

and

$\displaystyle\|w(t)\|_{\dot{H}^{1}}\leq\|w(0)\|_{\dot{H}^{1}}+C_{u_{0}}\int_{0}^{t}\|w\|_{\dot{H}^{1}\cap
L^{\infty}}ds.$ (3.19)

Combining (3.18), (3.19) and Gronwall’s inequality yield that

$\displaystyle\|w(t)\|_{\dot{H}^{1}\cap
L^{\infty}}\leq\|w(0)\|_{\dot{H}^{1}\cap
L^{\infty}}+C_{u_{0}}\int_{0}^{t}\|w\|_{\dot{H}^{1}\cap L^{\infty}}ds\leq
C_{u_{0}}\|w(0)\|_{\dot{H}^{1}\cap L^{\infty}}.$ (3.20)

Therefore, the uniqueness is obvious in view of (3.20). Moreover, an
interpolation argument ensures that the continuity with respect to the initial
data holds for the norm
$C([0,T];B^{\frac{3}{2}-\epsilon}_{\infty,2}\cap\dot{H}^{2-\epsilon})$ for
$\epsilon>0$ sufficient small. In fact, similar to the proof of [18], we can
raise the continuity with respect to the initial data until
$C([0,T];E^{2}_{2,2})$.

Step six. other cases.

Since we have prove the local well-posedness for (1.3) in
$C([0,T];E^{2}_{2,2})\cap C^{1}([0,T];\bar{E}^{2}_{2,2})$, other cases are
similar. In fact, we should only take some modifications in step three (step
six is similar).

Consider the Cauchy sequence $w^{n+1}:=u^{n+m+1}-u^{n+1}$:

$\frac{d}{dt}w^{n+1}+u^{n+1}w^{n+1}_{x}=-w^{n}u^{n+m+1}_{x}+\int_{\infty}^{x}w^{n+1}_{x}(u^{n+m}_{x}+u^{n}_{x})dz$
(3.21)

Similar to Step 2, it’s easy to deduce that $\\{u^{n}_{x}\\}$ is bounded in
$L^{q}$ if $u_{x0}\in L^{q}$. Then by virtue to Lemma 2.6 and the Holder
inequality, we have

$\displaystyle\|w^{n+1}(t)\|_{L^{\infty}}$ $\displaystyle\leq
C\int_{0}^{t}\|u^{n+m+1}_{x}\|_{L^{\infty}}\|w^{n+1}\|_{L^{\infty}}+\|u^{n+m}_{x}+u^{n}_{x}\|_{L^{q}}\|w^{n}\|_{\dot{W}^{1,p}}ds$
$\displaystyle\leq
C_{u_{0}}\int_{0}^{t}\|w^{n}\|_{L^{\infty}\cap\dot{W}^{1,p}}ds$ (3.22)

and

$\displaystyle\|w^{n+1}(t)\|_{\dot{W}^{1,p}}$ $\displaystyle\leq
C\int_{0}^{t}(\|u^{n+m+1}_{x}\|_{L^{\infty}}+\|u^{n}_{x}\|_{L^{\infty}})(\|w^{n}\|_{\dot{W}^{1,p}}+\|w^{n+1}\|_{\dot{W}^{1,p}})+(\|u^{n+m+1}_{xx}\|_{L^{p}})\|w^{n}\|_{L^{\infty}}ds$
$\displaystyle\leq
C_{u_{0}}\int_{0}^{t}\|w^{n}\|_{L^{\infty}\cap\dot{W}^{1,p}}ds,$ (3.23)

where the last inequality from the Gronwall’s inequality. Then Combining (3)
and (3), we get

$\displaystyle\|w^{n+1}(t)\|_{L^{\infty}\cap\dot{W}^{1,p}}\leq
C_{u_{0}}\int_{0}^{t}\|w^{n}\|_{L^{\infty}\cap\dot{W}^{1,p}}ds$ (3.24)

After some calculations we still get $\\{u^{n}\\}_{n=1}^{\infty}$ is a Cauchy
sequence in $L^{\infty}\cap\dot{W}^{1,p}$.

This complete the proof. ∎

######  Remark 3.3.

For $1\leq p\leq 2$, we have
$\dot{B}^{1}_{p,p}\cap\dot{B}^{2}_{p,p}\hookrightarrow\dot{W}^{1,q}$. But for
$p>2$, we have to consider it for an extra space $\dot{W}^{1,q}$ for the
initial data.

##  4 Blow-up and global existence

First we prove a conservation inequality for (1.3).

######  Lemma 4.1.

Let $u_{0}\in E^{s}_{2,2},s>\frac{5}{2}$ and $T^{*}$ be the maximal existence
time of the corresponding solution $u$ to (1.3), then we have

$\|u(t)\|_{\dot{H}_{1}}\leq\|u_{0}\|_{\dot{H}_{1}}.$

###### Proof.

Differentialing the equation (1.3) and multiplying with $2u_{x}$, we have

$\frac{d}{dt}u^{2}_{x}+(uu^{2}_{x})_{x}=0.$ (4.1)

This implies

$\displaystyle\frac{d}{dt}\|u(t)\|^{2}_{\dot{H}_{1}}=0.$

∎

Next we state a blow-up criterion for (1.3).

######  Lemma 4.2.

Let $u_{0}\in E^{s}_{2,2}=L^{\infty}\cap\dot{H}^{1}\cap\dot{H}^{s}$ with
$s>\frac{5}{2}$ being as in Theorem 3.2, and let $T^{*}$ be the maximal
existence time of the corresponding solution $u$ to (1.3). Then $u$ blows up
in finite time $T^{*}$ if and only if

$\int_{0}^{T^{*}}\|u_{x}(t^{\prime})\|_{L^{\infty}}dt^{\prime}=\infty.$

###### Proof.

Taking the $L^{\infty}$ norm to (1.3) both sides, we have

$\|u(t)\|_{L^{\infty}}\leq
C(\|u_{0}\|_{L^{\infty}}+\int_{0}^{t}\|u_{x}\|_{L^{\infty}}\|u\|_{L^{\infty}}+\|u_{0x}\|^{2}_{L^{2}}ds,$
(4.2)

Then differentiating the equation (1.3),

$\frac{d}{dt}u_{x}+uu_{xx}=-\frac{1}{2}u^{2}_{x}.$ (4.3)

By virtue to Lemma 2.6 and $u_{x}\in H^{s-1},s>\frac{5}{2}$, we get

$\|u_{x}(t)\|_{H^{s-1}}\leq
C(\|u_{0}\|_{H^{s-1}}+\int_{0}^{t}\|u_{x}\|_{L^{\infty}}\|u_{x}\|_{H^{s-1}}ds,$
(4.4)

Combining (4.2), (4.4) and the Gronwall inequality, we have

$\|u(t)\|_{E^{s}_{2,2}}\leq
Ce^{\int_{0}^{t}\|u_{x}\|_{L^{\infty}}ds}[(\|u_{0}\|_{E^{s}_{2,2}}+\|u_{0x}\|^{2}_{L^{2}}].$
(4.5)

If $T^{*}$ is finite, and
$\int_{0}^{T^{*}}\|u_{x}\|_{L^{\infty}}dt^{\prime}<\infty$, then $u\in
L^{\infty}([0,T^{*}];E^{s}_{2,2})$, which contradicts the assumption that
$T^{*}$ is the maximal existence time.

On the other hand, by Theorem 3.2 and the fact that
$E^{s}_{2,2}\hookrightarrow\dot{W}^{1,\infty}$, if
$\int_{0}^{T^{*}}\|u_{x}\|_{L^{\infty}}dt^{\prime}=\infty$, then $u$ must blow
up in finite time. ∎

######  Remark 4.3.

By the algebra interpolation, we can easily get a weaker blow-up criterion for
(1.3):

$\lim_{t\rightarrow T}\left\|u_{x}\right\|_{B^{0}_{\infty,\infty}}=\infty.$

Let us consider the ordinary differential equation:

$\left\\{\begin{array}[]{l}q_{t}(t,x)=u(t,q(t,x)),\quad t\in[0,T),\\\
q(0,x)=x,\quad x\in\mathbb{R}.\end{array}\right.$ (4.6)

If $u\in E^{s}_{2,2}$ with $s\geq 2$ being as in Theorem 3.2, then $u\in
C([0,T);C^{0,1})$. By the classical results in the theory of ordinary
differential equations, we can easily infer that (4.6) have a unique solution
$q\in C^{1}([0,T)\times\mathbb{R};\mathbb{R})$ such that the map $q(t,\cdot)$
is an increasing diffeomorphism of $\mathbb{R}$ with

$q_{x}(t,x)=\exp\Big{(}\int_{0}^{t}u(t^{\prime},q(t^{\prime},x)\Big{)}dt^{\prime}>0,\quad\forall(t,x)\in[0,T)\times\mathbb{R}.$

Now the following theorem shows that under particular condition for the
initial data, the corresponding solution of (1.3) will exist globally in time.

######  Theorem 4.4.

Let $u_{0}\in E^{s}_{2,2}$, $s>\frac{5}{2}$. Assume $u_{0xx}(x)\geq 0$ when
$x\leq x_{0}$, $u_{0xx}(x)\leq 0$ when $x\geq x_{0}$ for some
$x_{0}\in\mathbb{R}$, and $u_{0x}(x_{0})>0$. Then the corresponding solution
$u$ of (1.3) exists globally in time .

###### Proof.

Arguing by density, now we assume $s>\frac{7}{2}$. Differentiating (4.10), we
have

$\frac{d}{dt}u_{xx}+uu_{xxx}=-2u_{x}u_{xx}$ (4.7)

Then we have

$\frac{d}{dt}u_{xx}(t,q(t,x))=-2u_{x}u_{xx},$

Hence,

$u_{xx}(t,q(t,x))=u_{0xx}\exp\Big{(}\int_{0}^{t}-2u_{x}(t^{\prime},q(t^{\prime},x))dt^{\prime}\Big{)},$

which implies that $u_{xx}$ doesn’t change sign, so we can deduce that

$u_{xx}(t,x)\geq 0,\quad when\quad x\leq q(t,x_{0})\quad and\quad
u_{xx}(t,x)\leq 0,\quad when\quad x\geq q(t,x_{0}).$ (4.8)

Moreover, we have

$\frac{d}{dt}u_{x}(t,q(t,x))=-\frac{1}{2}u^{2}_{x}\leq 0.$ (4.9)

Using the fact that the flow $q(t,x)$ is a differmorphism and
$u_{0x}(x_{0})>0$, and by (4.8) we deduce that $q(t,x_{0})$ is the maximum
value point. (4.9) tells us $u_{x}(t)$ will decrease monotonly at the point
$q(t,x_{0})$ along the flow. Moreover, since $u_{x}(t)$ belongs to $H^{s-1}$,
it will decrease but will not be less than zero. Otherwise, it will contradict
with the decay of infinity in $H^{s}$.

As a result, we can deduce that the maximum point is at the initial point
$x_{0}$ such that

$\|u_{x}\|_{L^{\infty}}\leq\|u_{0x}\|_{L^{\infty}},$

which means the solution is global by Lemma 4.2 . ∎

######  Remark 4.5.

The initial datas satisfying the conditions of Theorem 4.4 are exist such as
$u_{0}(x)=\int_{-\infty}^{x}e^{-z^{2}}dz$. It’s easy to deduce that
$u_{0}(x)\in L^{\infty}\cap\dot{H}^{1}\cap\dot{H}^{2}$, $u_{0x}(0)>0$, and
$u_{0xx}(x)\geq 0$ when $x\leq 0$ and $u_{0xx}(x)\leq 0$ when $x\geq 0$.

We then shows that the corresponding solution of (1.3) will blow up by giving
some particular condition for the initial data.

######  Theorem 4.6.

Let $u_{0}\in E^{s}_{2,2}$, $s\geq 2$. Assume that there exists a point
$x_{0}$ such that $u_{0x}(x_{0})<0$, Then the corresponding solution $u$ of
(1.3) blows up in finite time.

###### Proof.

Arguing by density, now we assume $s>\frac{5}{2}$. From (4.10), we have

$\frac{d}{dt}u_{x}(t,q(t,x))=-\frac{1}{2}u^{2}_{x}.$ (4.10)

Solving the above equality, we finally get

$u_{x}(t,q(t,x))=\frac{2}{t+\frac{2}{u_{0x}}}$ (4.11)

As $u_{0x}(x_{0})<0$, we can easily deduce that the maximal time
$T<-\frac{2}{u_{0x}(x_{0})}$.

Therefore, from (4.11) we know $u_{x}(t)\rightarrow-\infty$ as $t\rightarrow
T$. By Lemma (4.2), the solution $u$ will blow up in finite time. ∎

##  5 ill-posedness

In this section, we are going to prove the norm inflation in $A\triangleq
L^{\infty}\cap\dot{B}^{\frac{1}{p}}_{p,r}\cap\dot{B}^{1+\frac{1}{p}}_{p,r}$
with $1\leq p\leq\infty,r>1$

######  Theorem 5.1.

Let $1<r\leq\infty,1\leq p\leq\infty$. For any $\epsilon>0$,there exists
$u_{0}\in H^{\infty}$ such that
(1)$\left\|u_{0}\right\|_{A}\leq\epsilon.$
(2)There exists a unique solution $u\in E^{\infty}_{p,r}$ with maximal
$T<\epsilon$.
(3)$\lim_{t\rightarrow
T}\left\|u_{x}\right\|_{B^{\frac{1}{p}}_{p,r}}\geq\lim_{t\rightarrow
T}\left\|u_{x}\right\|_{B^{0}{\infty,\infty}}=\infty.$

###### Proof.

We first define

$h(x)=\sum_{k\geq 1}^{\infty}\frac{h_{k}(x)}{2^{2k}k^{\frac{2}{1+r}}}$

where $h_{k}(\xi)=i2^{-k}\xi\varphi(2^{-k}\xi)$, and $\varphi(\xi)$ is even
non-negative, non-zero $C^{\infty}_{0}$ function in [22]. Similar to the proof
of [22], we can easily prove that

$\left\|h\right\|_{A}\leq\left\|h\right\|_{B^{1+\frac{1}{p}}_{p,r}}\leq
C,\quad\quad h^{\prime}(0)=-\infty$

For any $\epsilon>0$ small enough, let $u_{0}:=\frac{\epsilon
S_{N}h}{\left\|h\right\|_{B^{1+\frac{1}{p}}_{p,r}}}$, where $N$ is sufficient
large such that $u^{\prime}_{0}(0)<-2\epsilon^{-1}$. Then
$\left\|u_{0}\right\|_{A}\leq\epsilon$ and $u_{0}\in E^{\infty}_{p,r}$ for
fixed $N$.

Finally, by Theorem 3.2 and Theorem 4.2, we get a unique local solution $u\in
E^{\infty}_{p,r}$ and $u(t,x)$ blow up in finite time T, we deduce that

$\lim_{t\rightarrow
T}\left\|u_{x}\right\|_{B^{\frac{1}{p}}_{p,r}}\geq\lim_{t\rightarrow
T}\left\|u_{x}\right\|_{B^{0}_{\infty,\infty}}=\infty.$

Then we find the example:

$\left\|u_{0}-0\right\|_{A}\leq\epsilon,\quad\quad\left\|u(t)-0\right\|_{A}\rightarrow\infty$

This implies the ill-posedness of (1.3). ∎

######  Remark 5.2.

For $s=2,p=r=1$, we get the local well-posedness of (1.3) in
$E^{2}_{1,1}=L^{\infty}\cap\dot{B}^{1}_{1,1}\cap\dot{B}^{1}_{1,1}$, the above
theorem implies the solution will be ill-posedness in $E^{2}_{1,r},r>1$.

For $s=2,p=r=2$, we also get the local well-posedness of (1.3) in
$E^{2}_{2,2}=L^{\infty}\cap\dot{H}^{1}\cap\dot{H}^{2}$. However, the above
theorem only implies the solution will be ill-posedness in
$L^{\infty}\cap\dot{H}^{1}\cap\dot{H}^{\frac{3}{2}}$, we don’t whether the
solution will be well-posedness or ill-posedness in
$L^{\infty}\cap\dot{H}^{1}\cap\dot{H}^{s},\frac{3}{2}\leq s<2$.

##  6 Unique continuation

In this final section, we consider the unique continuation of $(\ref{eq1})$
with $g(t)=C\int_{-\infty}^{\infty}u_{x}^{2}(z)dz,\quad C\in\mathbb{R}$. We
first recall the equation :

$\left\\{\begin{array}[]{l}u_{x}+uu_{x}=\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz+C\int_{-\infty}^{\infty}u_{x}^{2}(z)dz.\\\
u(0,x)=u_{0}(x),\quad x\in\mathbb{R},\end{array}\right.$ (6.1)

######  Theorem 6.1.

Let $u(t,x)$ be a real strong solution of (1.3). If there exists a open set
$\Omega=(a,b)\times[t_{1},t_{2}]$, $(a,b\in\mathbb{R},t_{1},t_{2}\geq 0)$ such
that

$u(t,x)=0,\quad\quad(x,t)\in\Omega,$

and $u(t,x)$ meets one of the following cases:
1) For $C>0$
2) For $C=0$, $b=\infty$
3) For $-\frac{1}{2}<C<0$, $b=\infty$ or $a=-\infty$
4) For $C=-\frac{1}{2}$, $a=-\infty$
5) For $C<-\frac{1}{2}$
then $u\equiv 0$, $(x,t)\in\mathbb{R}^{+}\times\mathbb{R}$.

###### Proof.

1) For $C>0$, (1.3) can be wrote as

$u_{x}+uu_{x}=\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz+C\int_{-\infty}^{\infty}u_{x}^{2}(z)dz.$

From the hypothesis it follows that

$u_{x}+uu_{x}=0,\quad\quad(x,t)\in\Omega.$

This implies that

$\int_{-\infty}^{x}\frac{1}{2}u_{x}^{2}(z)dz+C\int_{-\infty}^{\infty}\frac{1}{2}u_{x}^{2}(z)dz=0,\quad(x,t)\in\Omega.$

As $\Omega=(a,b)\times[t_{1},t_{2}]$, we can easily get that
$u_{x}=0,\quad(x,t)\in\mathbb{R}\times[t_{1},t_{2}]$. The continuity of the
strong solution implies $u=0,\quad(x,t)\in\mathbb{R}\times[t_{1},t_{2}]$. Then
we get $u\equiv 0,\quad(x,t)\in\mathbb{R}\times\mathbb{R}^{+}$ by the
uniqueness.

Other cases are similar to 1), we omit it here. ∎

Acknowledgements. This work was partially supported by NNSFC (No. 11671407),
FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and
the key project of NSF of Guangdong province (No. 2016A03031104).

## References

  * [1] H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.
  * [2] A. Bressan, A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183(2) (2007), 215-239.
  * [3] A. Bressan, A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5(1) (2007), 1-27.
  * [4] A. Bressan, A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026.
  * [5] A. Bressan, H. Holden, X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pure. Appl., 94(1) (2010), 68-92.
  * [6] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier., 50 (2000), 321–362.
  * [7] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229–243.
  * [8] A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328.
  * [9] A. Constantin, J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475–504.
  * [10] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
  * [11] G, Chen, A. Bressan, Q, Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete. Continuous. Dynamical. Systems., 35(1) (2014), 25-42.
  * [12] H. He, Z. Yin, On a generalized Camassa-Holm equation with the flow generated by velocity and its gradient, Appl. Anal., (2016), 1-23.
  * [13] H. Holden, X. Raynaud, G. Katrin, Lipschitz metric for the periodic Camassa-Holm equation. J. Differ. Equ., 250(3) (2011), 1460-1492.
  * [14] H. Holden, X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier., 58 (2008), 945-988.
  * [15] A. N. W. Hone, V. Novikov, J. P. Wang, Generalizations of the short pulse equation, Lett. Math. Phys., 108 (2018), 927-947.
  * [16] J. K. Hunter, R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498–1521.
  * [17] J. K. Hunter, Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation. Phys. D., 79 (1994), 361-386.
  * [18] J. Li, Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differ. Equ., 261(11) (2016), 6125-6143.
  * [19] P. Olver, P. Rosenau, Tri-Hamiltonian duality between solitions and solitary wave solutions having compact support. Phys. Rev. E., 53 (1996), 1900-1906.
  * [20] R. Beals, D. Sattinger, J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equations, Appl. Anal., 78 (2001), 255-269.
  * [21] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differ. Equ., 192(2) (2003), 429-444.
  * [22] Z. Guo, X Liu, L Molinet, Z. Yin. Ill-posedness of the Camassa-Holm and related equations in the critical space, J. Differ. Equ., 266 (2019), 1698-1707.
  * [23] Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
  * [24] Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283.