110 |  G24.33$+$0.1M1 | 18:35:08.132 |  $-$07:35:04.176 |  |  00.438$\pm$0.007 |  000.434$\pm$0.007 | ¡5400 |  |  00.769$\pm$0.029 |  000.905$\pm$0.034 |  $<$1600 | iC/iK |  $+$0.55$\pm$0.09†
111 |  | 18:35:13.569 |  $-$07:38:20.379 |  |  00.382$\pm$0.014 |  000.659$\pm$0.024 |  ¡2700 |  | … | … | … | iC/oK | …
112 |  | 18:35:33.971 |  $-$07:37:34.491 |  |  01.630$\pm$0.078 |  004.685$\pm$0.228 |  ¡7400 |  | … | … | … | iC/oK | …
113 |  | 18:35:10.888 |  $-$07:34:22.080 |  |  00.198$\pm$0.007 |  000.278$\pm$0.011 |  ¡3200 |  | … | … | … | iC/iK |  $<+$0.07
114 |  | 18:35:02.821 |  $-$07:31:20.729 |  |  00.290$\pm$0.005 |  000.328$\pm$0.005 |  ¡2000 |  | … | … | … | iC/oK | …
115 |  | 18:34:48.656 |  $-$07:46:40.369 |  |  00.151$\pm$0.006 |  000.225$\pm$0.009 |  ¡3700 |  | … | … | … | oC | …
116 |  | 18:35:56.006 |  $-$07:27:23.278 |  |  00.196$\pm$0.009 |  000.289$\pm$0.015 |  ¡2400 |  | … | … | … | oC | …
117 |  G24.60$+$0.1M1 | 18:36:12.517 |  $-$07:12:09.922 |  |  00.437$\pm$0.015 |  000.526$\pm$0.018 |  ¡1300 |  | … | … | … | oC | …
118 |  | 18:36:12.607 |  $-$07:12:14.144 |  |  00.420$\pm$0.022 |  000.983$\pm$0.052 |  ¡5700 |  | … | … | … | oC | …
119 |  | 18:35:40.677 |  $-$07:22:05.797 |  |  00.305$\pm$0.016 |  001.107$\pm$0.059 |  ¡8400 |  | … | … | … | iC/oK | …
120 |  | 18:35:40.739 |  $-$07:22:00.863 |  |  00.284$\pm$0.009 |  000.457$\pm$0.014 |  ¡3000 |  | … | … | … | iC/oK | …
121 |  | 18:35:40.869 |  $-$07:21:57.887 |  |  00.286$\pm$0.020 |  001.705$\pm$0.126 |  ¡109000 |  | … | … | … | iC/oK | …
122 |  | 18:36:05.589 |  $-$07:31:21.640 |  |  00.349$\pm$0.036 |  003.790$\pm$0.390 |  ¡162000 |  | … | … | … | oC | …
123 |  | 18:35:53.123 |  $-$07:14:20.492 |  |  01.058$\pm$0.011 |  001.164$\pm$0.012 | ¡5500 |  | … | … | … | iC/oK | …
124 |  | 18:35:16.807 |  $-$07:05:08.357 |  |  00.239$\pm$0.011 |  000.400$\pm$0.019 |  ¡3800 |  | … | … | … | oC | …
125 |  | 18:35:41.070 |  $-$07:16:41.062 |  |  00.140$\pm$0.004 |  000.123$\pm$0.004 | ¡4900 |  | … | … | … | iC/iK | …
126 |  | 18:35:43.769 |  $-$07:19:26.186 |  |  00.114$\pm$0.003 |  000.105$\pm$0.002 | ¡5000 |  | … | … | … | iC/iK |  $<+$0.96
127 |  | 18:35:03.755 |  $-$07:26:00.913 |  |  00.146$\pm$0.012 |  000.529$\pm$0.044 |  ¡8400 |  | … | … | … | oC | …
128 |  | 18:36:18.116 |  $-$07:08:50.104 |  |  00.128$\pm$0.003 |  000.705$\pm$0.018 |  ¡110000 |  | … | … | … | oC | …
129 |  | 18:35:40.400 |  $-$07:19:28.740 |  |  00.088$\pm$0.005 |  000.157$\pm$0.009 |  ¡4500 |  | … | … | … | iC/iK |  $<+$0.18
130 |  | 18:34:59.859 |  $-$07:26:39.367 |  |  00.078$\pm$0.002 |  000.110$\pm$0.003 |  ¡3000 |  | … | … | … | oC | …
131 |  | 18:34:39.433 |  $-$07:02:39.152 |  |  00.068$\pm$0.002 |  000.081$\pm$0.003 | ¡5700 |  | … | … | … | oC | …
132 |  G24.60$+$0.1M2 | 18:35:47.199 |  $-$07:12:59.443 |  |  00.300$\pm$0.008 |  000.543$\pm$0.015 |  ¡3800 |  | … | … | … | iC/oK | …
133 |  | 18:35:40.955 |  $-$07:21:56.177 |  |  00.275$\pm$0.010 |  000.992$\pm$0.036 |  ¡7600 |  | … | … | … | iC/oK | …
134 |  | 18:35:56.070 |  $-$07:27:23.872 |  |  00.133$\pm$0.015 |  000.243$\pm$0.028 | ¡6500 |  | … | … | … | oC | …
135 |  | 18:35:29.951 |  $-$07:27:46.840 |  |  00.064$\pm$0.003 |  000.077$\pm$0.003 |  ¡1900 |  | … | … | … | oC | …
136 |  | 18:35:40.124 |  $-$07:18:37.396 |  | … | … | … |  |  00.821$\pm$0.110 |  000.853$\pm$0.116 |  $<$2100 | iC/iK |  $>+$1.67
137 |  G34.43$+$0.2M3 | 18:53:18.845 |  $+$01:14:59.320 |  |  04.070$\pm$0.280 |  018.400$\pm$1.300 |  ¡7000 |  |  00.242$\pm$0.022 |  001.270$\pm$0.120 |  ¡2000 | oC | …
138 |  | 18:53:18.672 |  $+$01:24:47.737 |  |  06.046$\pm$0.080 |  007.393$\pm$0.097 |  ¡1600 |  | … | … | … | iC/oK | …
139 |  | 18:53:18.020 |  $+$01:25:25.608 |  |  00.439$\pm$0.017 |  000.729$\pm$0.029 |  ¡2100 |  | … | … | … | iC/oK | …
140 |  | 18:53:08.320 |  $+$01:29:33.994 |  |  00.858$\pm$0.013 |  000.820$\pm$0.012 |  ¡3000 |  | … | … | … | iC/oK | …
141 |  | 18:53:35.990 |  $+$01:35:18.770 |  |  02.290$\pm$0.147 |  010.435$\pm$0.672 | ¡5600 |  | … | … | … | iC/oK | …
142 |  | 18:54:14.014 |  $+$01:19:18.429 |  |  00.089$\pm$0.003 |  000.148$\pm$0.005 |  ¡7100 |  | … | … | … | oC | …
143 |  IRAS 19095$+$0930 | 19:11:53.997 |  $+$09:35:50.404 |  |  30.580$\pm$0.860 |  040.400$\pm$1.100 |  ¡1900 |  |  90.545$\pm$2.101 |  126.364$\pm$3.001 |  ¡0660 | iC/iK |  $+$0.87$\pm$0.08†
144 |  | 19:11:54.369 |  $+$09:35:49.254 |  |  04.211$\pm$0.240 |  015.565$\pm$0.890 |  ¡7300 |  | … | … | … | iC/iK |  $<-$2.97
145 |  | 19:12:00.546 |  $+$09:36:24.232 |  |  02.847$\pm$0.023 |  002.838$\pm$0.023 | ¡4400 |  | … | … | … | iC/iK |  $<+$0.32
146 |  | 19:11:46.436 |  $+$09:37:03.679 |  |  01.927$\pm$0.025 |  002.048$\pm$0.027 | ¡4500 |  | … | … | … | iC/oK | …
Table 5: continued. 999 $b$$b$footnotetext: Flux density, intensity peak and deconvolved source size for the sources detected at 1.3 cm in the K band images. The fluxes are corrected by the primary beam response of the antennas, except for sources located outside the primary beam (listed either as ‘iC/oK’ or ‘oC’, see Sect. 3.1 for more details about this classification). Table 6:  Observed and deconvolved angular sizes for the radio continuum sources detected in this work (see Table LABEL:t:catalogue) |  | Observed source size |  | Deconvolved diameter
---|---|---|---|---
|  | Major axis | Minor axis | PA |  | Major axis | Minor axis | PA
ID | Region | (″) | (″) | (°) |  | (″) | (″) | (°)
Observed and deconvolved source sizes for C band detections
1 |  IRAS 05358$+$3543 |  01.269$\pm$0.009 |  01.239$\pm$0.009 |  053.0$\pm$3.8 |  | … | … | …
2 |  |  01.532$\pm$0.034 |  01.402$\pm$0.034 |  106.7$\pm$3.4 |  |  00.894$\pm$0.063 |  00.633$\pm$0.085 |  114.9$\pm$9.9
3 |  |  01.113$\pm$0.044 |  01.001$\pm$0.043 |  155.4$\pm$5.2 |  | … | … | …
4 |  |  02.411$\pm$0.058 |  01.455$\pm$0.057 |  120.4$\pm$1.1 |  |  02.069$\pm$0.069 |  00.730$\pm$0.124 |  121.1$\pm$1.2
5 |  |  01.667$\pm$0.041 |  01.409$\pm$0.041 |  104.5$\pm$2.0 |  |  01.105$\pm$0.065 |  00.654$\pm$0.098 |  108.5$\pm$3.8
6 |  |  01.412$\pm$0.028 |  01.349$\pm$0.027 |  000.1$\pm$7.5 |  |  00.690$\pm$0.066 |  00.472$\pm$0.094 |  169.0$\pm$9.0
7 |  |  01.520$\pm$0.046 |  01.424$\pm$0.044 |  145.4$\pm$6.4 |  |  00.899$\pm$0.082 |  00.644$\pm$0.106 |  147.1$\pm$6.4
8 |  |  01.330$\pm$0.035 |  01.286$\pm$0.034 |  168.0$\pm$9.0 |  |  00.512$\pm$0.758 |  00.215$\pm$1.011 |  160.0$\pm$9.0
9 |  |  01.357$\pm$0.012 |  01.255$\pm$0.012 |  076.4$\pm$1.4 |  |  00.490$\pm$0.035 |  00.249$\pm$0.076 |  085.0$\pm$9.0
10 |  |  01.918$\pm$0.083 |  01.502$\pm$0.086 |  065.3$\pm$3.1 |  |  01.440$\pm$0.110 |  00.870$\pm$0.160 |  065.6$\pm$3.8
11 |  |  01.089$\pm$0.070 |  01.065$\pm$0.068 |  135.0$\pm$9.0 |  | … | … | …
12 |  G189.78$+$0.34 |  01.673$\pm$0.041 |  01.397$\pm$0.048 |  012.0$\pm$2.2 |  |  01.095$\pm$0.070 |  00.860$\pm$0.089 |  000.5$\pm$9.6
13 |  |  01.578$\pm$0.041 |  01.249$\pm$0.048 |  014.6$\pm$1.8 |  |  00.934$\pm$0.074 |  00.602$\pm$0.115 |  006.7$\pm$9.5
14 |  |  01.781$\pm$0.072 |  01.441$\pm$0.083 |  010.1$\pm$3.9 |  |  01.250$\pm$0.110 |  00.930$\pm$0.150 |  001.7$\pm$9.4
15 |  |  02.461$\pm$0.073 |  01.688$\pm$0.050 |  158.3$\pm$2.0 |  |  02.161$\pm$0.088 |  01.194$\pm$0.079 |  154.2$\pm$2.7
16 |  |  01.920$\pm$0.050 |  01.493$\pm$0.050 |  156.9$\pm$1.8 |  |  01.525$\pm$0.068 |  00.880$\pm$0.096 |  148.3$\pm$3.9
17 |  |  01.014$\pm$0.052 |  00.892$\pm$0.061 |  018.6$\pm$6.4 |  | … | … | …
18 |  |  01.598$\pm$0.078 |  01.254$\pm$0.092 |  021.7$\pm$4.9 |  |  00.958$\pm$0.136 |  00.629$\pm$0.226 |  020.0$\pm$9.0
19 |  |  01.959$\pm$0.091 |  01.148$\pm$0.100 |  174.7$\pm$2.0 |  |  01.524$\pm$0.091 |  00.136$\pm$0.100 |  169.9$\pm$2.0
20 |  |  01.382$\pm$0.071 |  00.961$\pm$0.083 |  016.5$\pm$2.4 |  | … | … | …
21 |  G192.58$-$0.04 |  01.741$\pm$0.046 |  01.451$\pm$0.054 |  025.8$\pm$2.2 |  |  01.033$\pm$0.082 |  00.827$\pm$0.106 |  032.0$\pm$9.0
22 |  |  02.524$\pm$0.064 |  01.700$\pm$0.029 |  059.1$\pm$1.5 |  |  02.152$\pm$0.078 |  01.116$\pm$0.052 |  063.7$\pm$2.1
23 |  |  01.454$\pm$0.036 |  01.162$\pm$0.043 |  030.3$\pm$1.8 |  | … | … | …
24 |  |  01.206$\pm$0.066 |  01.073$\pm$0.072 |  170.4$\pm$6.4 |  | … | … | …
25 |  |  02.903$\pm$0.050 |  02.375$\pm$0.038 |  148.1$\pm$2.4 |  |  02.614$\pm$0.060 |  01.961$\pm$0.052 |  143.0$\pm$3.1
26 |  |  01.745$\pm$0.050 |  01.247$\pm$0.042 |  028.1$\pm$1.9 |  |  01.043$\pm$0.064 |  00.360$\pm$0.221 |  032.0$\pm$4.4
27 |  G18.18$-$0.30 |  03.480$\pm$0.102 |  02.620$\pm$0.077 |  047.0$\pm$1.4 |  |  03.270$\pm$0.110 |  02.030$\pm$0.110 |  053.9$\pm$2.4
28 |  |  01.884$\pm$0.040 |  01.550$\pm$0.064 |  174.8$\pm$2.4 |  |  01.180$\pm$0.098 |  00.657$\pm$0.172 |  061.0$\pm$9.0
29 |  |  02.210$\pm$0.025 |  01.604$\pm$0.033 |  011.5$\pm$0.8 |  |  01.629$\pm$0.042 |  00.820$\pm$0.085 |  037.9$\pm$3.1
30 |  |  01.962$\pm$0.017 |  01.419$\pm$0.028 |  165.8$\pm$0.8 |  |  00.971$\pm$0.050 |  00.890$\pm$0.053 |  050.0$\pm$9.0
31 |  |  03.244$\pm$0.057 |  02.909$\pm$0.052 |  151.8$\pm$4.5 |  |  02.803$\pm$0.090 |  02.645$\pm$0.091 |  117.0$\pm$9.0
32 |  |  01.926$\pm$0.016 |  01.591$\pm$0.024 |  179.5$\pm$0.7 |  |  01.267$\pm$0.034 |  00.711$\pm$0.058 |  057.4$\pm$3.7
33 |  |  01.637$\pm$0.011 |  01.047$\pm$0.019 |  166.6$\pm$0.3 |  | … | … | …
34 |  |  01.714$\pm$0.030 |  01.148$\pm$0.049 |  161.5$\pm$1.0 |  | … | … | …
35 |  |  02.876$\pm$0.108 |  02.066$\pm$0.072 |  056.7$\pm$1.5 |  |  02.660$\pm$0.120 |  01.160$\pm$0.140 |  062.3$\pm$2.0
36 |  |  02.114$\pm$0.037 |  01.225$\pm$0.061 |  157.9$\pm$1.0 |  |  01.224$\pm$0.073 |  00.584$\pm$0.164 |  147.5$\pm$6.1
37 |  |  03.407$\pm$0.061 |  02.362$\pm$0.099 |  156.2$\pm$1.4 |  |  02.938$\pm$0.074 |  02.105$\pm$0.117 |  152.7$\pm$3.1
38 |  |  03.376$\pm$0.103 |  02.275$\pm$0.068 |  132.8$\pm$1.9 |  |  03.000$\pm$0.130 |  01.850$\pm$0.110 |  123.8$\pm$3.9
39 |  IRAS 18223$-$1243 |  02.267$\pm$0.027 |  01.865$\pm$0.034 |  018.1$\pm$0.8 |  |  01.715$\pm$0.043 |  00.882$\pm$0.088 |  054.0$\pm$3.4
40 |  |  02.167$\pm$0.030 |  01.550$\pm$0.048 |  172.3$\pm$0.7 |  |  01.163$\pm$0.073 |  00.920$\pm$0.096 |  033.0$\pm$9.0
41 |  |  02.230$\pm$0.044 |  01.423$\pm$0.072 |  166.4$\pm$1.0 |  |  01.178$\pm$0.085 |  00.844$\pm$0.132 |  168.0$\pm$9.0
42 |  |  02.064$\pm$0.021 |  01.461$\pm$0.034 |  175.2$\pm$0.5 |  |  01.053$\pm$0.055 |  00.622$\pm$0.099 |  041.7$\pm$7.7
43 |  |  03.002$\pm$0.060 |  02.684$\pm$0.098 |  161.3$\pm$2.7 |  |  02.440$\pm$0.120 |  02.320$\pm$0.100 |  091.0$\pm$9.0
44 |  |  03.271$\pm$0.056 |  02.803$\pm$0.078 |  142.1$\pm$1.7 |  |  02.810$\pm$0.081 |  02.401$\pm$0.104 |  116.0$\pm$9.0
45 |  |  02.538$\pm$0.046 |  01.970$\pm$0.075 |  158.5$\pm$1.5 |  |  01.748$\pm$0.088 |  01.538$\pm$0.113 |  129.0$\pm$9.0
46 |  |  02.548$\pm$1.973 |  01.973$\pm$0.091 |  157.0$\pm$7.0 |  | … | … | …
47 |  |  05.050$\pm$0.250 |  04.120$\pm$0.200 |  143.0$\pm$9.0 |  | … | … | …
48a |  IRAS 18228$-$1312 |  06.450$\pm$0.144 |  03.501$\pm$0.070 |  004.3$\pm$1.2 |  |  06.189$\pm$0.153 |  03.268$\pm$0.080 |  005.7$\pm$1.5
48b |  |  07.250$\pm$0.270 |  05.940$\pm$0.220 |  046.1$\pm$7.4 |  |  07.120$\pm$0.290 |  05.670$\pm$0.250 |  049.0$\pm$8.0
48c |  |  04.610$\pm$0.340 |  03.640$\pm$0.250 |  065.0$\pm$9.0 |  |  04.450$\pm$0.370 |  03.130$\pm$0.320 |  067.0$\pm$9.0
49 |  |  01.784$\pm$0.025 |  01.160$\pm$0.040 |  163.8$\pm$0.7 |  | … | … | …
50 |  |  11.100$\pm$0.080 |  08.271$\pm$0.107 |  140.4$\pm$0.5 |  |  10.960$\pm$0.082 |  08.164$\pm$0.109 |  139.5$\pm$0.6
51 |  G19.27$+$0.1M1 |  04.790$\pm$0.110 |  02.980$\pm$0.130 |  167.3$\pm$1.7 |  |  04.380$\pm$0.120 |  02.730$\pm$0.150 |  167.9$\pm$2.2
52 |  |  02.071$\pm$0.050 |  01.231$\pm$0.082 |  163.8$\pm$1.7 |  |  00.694$\pm$1.257 |  00.317$\pm$0.872 |  161.0$\pm$9.0
53 |  |  02.018$\pm$0.021 |  01.218$\pm$0.034 |  164.9$\pm$0.7 |  |  00.518$\pm$1.433 |  00.258$\pm$0.931 |  173.0$\pm$9.0
54 |  |  02.060$\pm$0.028 |  01.387$\pm$0.042 |  171.9$\pm$1.3 |  |  00.889$\pm$1.063 |  00.391$\pm$0.797 |  036.0$\pm$9.0
55 |  |  04.480$\pm$0.140 |  03.170$\pm$0.110 |  095.3$\pm$2.9 |  |  04.290$\pm$0.150 |  00.550$\pm$0.150 |  091.0$\pm$9.0
56 |  |  08.700$\pm$0.260 |  06.540$\pm$0.260 |  100.9$\pm$4.2 |  |  08.590$\pm$0.280 |  06.280$\pm$0.280 |  099.3$\pm$4.4
57 |  |  03.057$\pm$0.094 |  02.282$\pm$0.092 |  141.4$\pm$3.5 |  |  02.480$\pm$0.140 |  01.780$\pm$0.170 |  124.0$\pm$9.0
58 |  |  02.447$\pm$0.037 |  01.724$\pm$0.059 |  175.3$\pm$1.3 |  |  01.571$\pm$0.077 |  01.125$\pm$0.116 |  019.6$\pm$9.9
59 |  |  01.951$\pm$0.033 |  01.147$\pm$0.053 |  159.8$\pm$1.2 |  | … | … | …
60 |  |  02.300$\pm$0.140 |  01.430$\pm$0.120 |  170.1$\pm$3.6 |  |  01.250$\pm$0.310 |  00.730$\pm$0.480 |  004.2$\pm$9.8
61 |  |  02.950$\pm$0.100 |  01.900$\pm$0.100 |  152.8$\pm$2.5 |  |  02.250$\pm$0.150 |  01.430$\pm$0.170 |  143.8$\pm$8.4
63 |  IRAS 18236$-$1205 |  00.485$\pm$0.019 |  00.393$\pm$0.021 |  134.6$\pm$7.2 |  | … | … | …
64 |  |  02.229$\pm$0.061 |  01.148$\pm$0.079 |  170.1$\pm$1.5 |  | … | … | …
65 |  |  02.033$\pm$0.012 |  01.112$\pm$0.017 |  163.6$\pm$0.3 |  | … | … | …
66 |  |  01.785$\pm$0.044 |  01.121$\pm$0.042 |  156.1$\pm$1.6 |  | … | … | …
67 |  |  04.068$\pm$1.886 |  01.886$\pm$0.093 |  116.0$\pm$2.0 |  | … | … | …
68 |  |  02.633$\pm$0.156 |  01.893$\pm$0.092 |  138.0$\pm$6.0 |  | … | … | …
69 |  |  06.040$\pm$0.820 |  02.180$\pm$0.210 |  003.0$\pm$3.0 |  | … | … | …
70 |  |  01.840$\pm$0.170 |  01.400$\pm$0.100 |  126.0$\pm$9.0 |  | … | … | …
71 |  |  03.193$\pm$0.223 |  01.823$\pm$0.093 |  159.0$\pm$4.0 |  | … | … | …
72 |  G23.60$+$0.0M1 |  05.110$\pm$0.210 |  04.960$\pm$0.210 |  114.0$\pm$9.0 |  |  04.930$\pm$0.290 |  04.660$\pm$0.300 |  086.0$\pm$9.0
73 |  |  01.770$\pm$0.029 |  01.175$\pm$0.035 |  160.1$\pm$1.4 |  | … | … | …
74 |  |  01.793$\pm$0.021 |  01.236$\pm$0.025 |  160.3$\pm$1.0 |  | … | … | …
75 |  |  01.884$\pm$0.020 |  01.198$\pm$0.026 |  156.3$\pm$0.8 |  |  00.444$\pm$1.410 |  00.290$\pm$0.842 |  105.0$\pm$9.0
76 |  |  01.926$\pm$0.019 |  01.230$\pm$0.032 |  157.3$\pm$0.8 |  |  00.541$\pm$0.083 |  00.475$\pm$0.117 |  141.0$\pm$9.0
77 |  |  01.975$\pm$0.017 |  01.372$\pm$0.027 |  162.3$\pm$0.9 |  |  00.847$\pm$0.061 |  00.602$\pm$0.087 |  035.0$\pm$9.0
78 |  |  02.285$\pm$0.041 |  01.856$\pm$0.057 |  152.2$\pm$3.1 |  |  01.497$\pm$0.095 |  01.312$\pm$0.111 |  088.0$\pm$9.0
79 |  |  02.672$\pm$0.090 |  01.760$\pm$0.116 |  174.9$\pm$2.6 |  |  02.010$\pm$0.140 |  01.220$\pm$0.220 |  008.9$\pm$9.8
80 |  |  01.817$\pm$0.065 |  01.405$\pm$0.084 |  173.7$\pm$4.9 |  | … | … | …
81 |  |  01.828$\pm$0.051 |  01.418$\pm$0.080 |  169.0$\pm$4.0 |  | … | … | …
82 |  |  01.762$\pm$0.033 |  01.324$\pm$0.054 |  152.6$\pm$2.3 |  | … | … | …
83 |  IRAS 18316$-$0602 |  01.987$\pm$0.041 |  01.376$\pm$0.060 |  174.1$\pm$1.9 |  |  01.181$\pm$0.096 |  00.592$\pm$0.228 |  022.5$\pm$9.8
84 |  |  01.741$\pm$0.018 |  01.230$\pm$0.028 |  163.2$\pm$1.1 |  |  00.620$\pm$1.085 |  00.251$\pm$0.838 |  046.0$\pm$9.0
85 |  |  01.759$\pm$0.014 |  01.253$\pm$0.021 |  165.1$\pm$0.9 |  |  00.696$\pm$0.053 |  00.283$\pm$0.184 |  041.5$\pm$7.6
86 |  |  01.683$\pm$0.021 |  01.250$\pm$0.031 |  163.2$\pm$1.7 |  | … | … | …
87 |  |  01.685$\pm$0.021 |  01.215$\pm$0.032 |  159.8$\pm$1.4 |  | … | … | …
88 |  |  04.688$\pm$0.090 |  02.687$\pm$0.094 |  013.0$\pm$1.1 |  |  04.434$\pm$0.099 |  02.333$\pm$0.117 |  016.2$\pm$1.5
89 |  |  02.250$\pm$0.100 |  01.690$\pm$0.110 |  118.4$\pm$4.7 |  |  01.790$\pm$0.160 |  00.800$\pm$0.400 |  098.0$\pm$9.0
90 |  |  01.709$\pm$0.050 |  01.240$\pm$0.074 |  145.2$\pm$3.2 |  | … | … | …
91 |  |  01.521$\pm$0.081 |  01.376$\pm$0.099 |  172.0$\pm$9.0 |  | … | … | …
92 |  |  02.140$\pm$0.110 |  01.500$\pm$0.110 |  145.9$\pm$4.7 |  |  01.380$\pm$0.230 |  00.910$\pm$0.330 |  124.0$\pm$9.0
93 |  |  02.474$\pm$0.119 |  01.509$\pm$0.075 |  173.9$\pm$2.4 |  |  01.840$\pm$0.180 |  00.950$\pm$0.170 |  004.0$\pm$7.9
96 |  G24.08$+$0.0M2 |  01.977$\pm$0.023 |  01.317$\pm$0.036 |  150.6$\pm$1.0 |  |  00.942$\pm$0.068 |  00.487$\pm$0.151 |  118.6$\pm$8.7
97 |  |  01.935$\pm$0.026 |  01.402$\pm$0.039 |  171.8$\pm$1.5 |  |  01.008$\pm$0.078 |  00.418$\pm$0.265 |  037.1$\pm$7.8
98 |  |  02.469$\pm$0.022 |  01.492$\pm$0.022 |  174.4$\pm$0.6 |  |  01.759$\pm$0.035 |  00.842$\pm$0.052 |  006.8$\pm$1.8
99 |  |  01.940$\pm$0.019 |  01.464$\pm$0.030 |  161.5$\pm$1.2 |  |  00.942$\pm$0.054 |  00.708$\pm$0.068 |  059.0$\pm$9.0
100 |  |  02.967$\pm$0.039 |  01.410$\pm$0.050 |  159.1$\pm$0.7 |  |  02.357$\pm$0.050 |  00.846$\pm$0.086 |  159.1$\pm$1.0
101 |  |  02.471$\pm$0.101 |  01.714$\pm$0.054 |  022.8$\pm$3.0 |  |  02.020$\pm$0.140 |  00.660$\pm$0.280 |  039.1$\pm$5.2
102 |  |  01.907$\pm$0.016 |  01.358$\pm$0.022 |  161.6$\pm$0.9 |  |  00.776$\pm$0.049 |  00.599$\pm$0.070 |  050.0$\pm$9.0
103 |  |  02.146$\pm$0.049 |  01.399$\pm$0.053 |  153.8$\pm$1.7 |  |  01.190$\pm$0.100 |  00.790$\pm$0.130 |  140.0$\pm$9.0
104 |  |  02.180$\pm$0.150 |  01.540$\pm$0.110 |  159.7$\pm$5.7 |  |  01.230$\pm$0.270 |  01.050$\pm$0.270 |  162.0$\pm$9.0
105 |  |  03.350$\pm$0.270 |  02.830$\pm$0.210 |  162.0$\pm$9.0 |  | … | … | …
106 |  |  26.310$\pm$1.400 |  19.000$\pm$0.530 |  069.7$\pm$4.3 |  |  26.280$\pm$1.400 |  18.910$\pm$0.540 |  069.7$\pm$4.3
107 |  |  01.369$\pm$0.138 |  01.020$\pm$0.072 |  160.0$\pm$9.6 |  | … | … | …
108 |  |  01.540$\pm$1.289 |  01.289$\pm$0.069 |  010.0$\pm$9.1 |  | … | … | …
109 |  |  00.940$\pm$0.130 |  00.940$\pm$0.130 |  168.0$\pm$7.7 |  | … | … | …
110 |  G24.33$+$0.1M1 |  01.661$\pm$0.021 |  01.259$\pm$0.029 |  166.9$\pm$0.7 |  | … | … | …
111 |  |  02.114$\pm$0.064 |  01.722$\pm$0.047 |  078.2$\pm$1.8 |  |  01.708$\pm$0.081 |  00.315$\pm$0.931 |  078.0$\pm$9.0
112 |  |  02.571$\pm$0.064 |  02.361$\pm$0.081 |  147.2$\pm$4.0 |  |  02.070$\pm$0.110 |  01.860$\pm$0.120 |  103.0$\pm$9.0
113 |  |  01.957$\pm$0.051 |  01.511$\pm$0.066 |  150.4$\pm$1.8 |  |  01.120$\pm$0.110 |  00.650$\pm$0.220 |  121.0$\pm$9.0
114 |  |  01.821$\pm$0.022 |  01.314$\pm$0.030 |  167.3$\pm$0.7 |  |  00.670$\pm$0.064 |  00.414$\pm$0.114 |  170.0$\pm$9.0
115 |  |  01.976$\pm$0.051 |  01.584$\pm$0.068 |  155.8$\pm$2.0 |  |  01.120$\pm$0.120 |  00.860$\pm$0.160 |  121.0$\pm$9.0
116 |  |  01.861$\pm$0.079 |  01.673$\pm$0.072 |  112.4$\pm$4.5 |  |  01.321$\pm$0.079 |  00.307$\pm$0.072 |  087.9$\pm$4.5
117 |  G24.60$+$0.1M1 |  01.681$\pm$0.041 |  01.456$\pm$0.056 |  168.5$\pm$3.9 |  |  00.800$\pm$0.873 |  00.170$\pm$1.500 |  082.0$\pm$9.0
118 |  |  02.304$\pm$0.086 |  02.068$\pm$0.067 |  060.9$\pm$6.6 |  |  01.940$\pm$0.110 |  01.240$\pm$0.130 |  071.6$\pm$6.6
119 |  |  03.087$\pm$0.074 |  02.396$\pm$0.081 |  132.6$\pm$2.5 |  |  02.708$\pm$0.096 |  01.914$\pm$0.118 |  122.5$\pm$5.1
120 |  |  02.232$\pm$0.048 |  01.467$\pm$0.052 |  020.8$\pm$1.5 |  |  01.636$\pm$0.076 |  00.426$\pm$0.791 |  035.0$\pm$9.0
121 |  |  05.480$\pm$0.200 |  02.210$\pm$0.110 |  034.4$\pm$1.5 |  |  05.280$\pm$0.210 |  01.650$\pm$0.160 |  035.9$\pm$1.7
122 |  |  06.720$\pm$0.150 |  03.290$\pm$0.150 |  038.5$\pm$1.3 |  |  06.570$\pm$0.160 |  02.930$\pm$0.170 |  039.6$\pm$1.4
123 |  |  01.856$\pm$0.013 |  01.205$\pm$0.018 |  163.1$\pm$0.5 |  | … | … | …
124 |  |  01.902$\pm$0.066 |  01.788$\pm$0.073 |  025.0$\pm$9.0 |  |  01.380$\pm$0.130 |  00.800$\pm$0.250 |  071.0$\pm$9.0
125 |  |  01.594$\pm$0.041 |  01.122$\pm$0.056 |  168.9$\pm$2.3 |  | … | … | …
126 |  |  01.708$\pm$0.042 |  01.097$\pm$0.044 |  162.8$\pm$1.8 |  | … | … | …
127 |  |  02.910$\pm$0.120 |  02.530$\pm$0.130 |  132.0$\pm$7.6 |  |  02.530$\pm$0.170 |  02.050$\pm$0.200 |  114.0$\pm$9.0
128 |  |  04.754$\pm$0.072 |  02.358$\pm$0.042 |  005.9$\pm$0.6 |  |  04.462$\pm$0.078 |  01.992$\pm$0.052 |  007.2$\pm$0.8
129 |  |  02.374$\pm$0.071 |  01.524$\pm$0.096 |  176.8$\pm$2.1 |  |  01.690$\pm$0.110 |  00.900$\pm$0.190 |  001.1$\pm$5.6
130 |  |  01.908$\pm$0.037 |  01.501$\pm$0.047 |  151.2$\pm$3.0 |  |  01.115$\pm$0.097 |  00.612$\pm$0.201 |  118.0$\pm$9.0
131 |  |  02.076$\pm$0.077 |  01.161$\pm$0.061 |  165.9$\pm$2.1 |  | … | … | …
132 |  G24.60$+$0.01M2 |  02.003$\pm$0.081 |  01.537$\pm$0.046 |  137.2$\pm$3.8 |  |  01.400$\pm$0.130 |  00.790$\pm$0.220 |  103.0$\pm$9.0
133 |  |  02.778$\pm$0.097 |  02.206$\pm$0.101 |  000.1$\pm$5.5 |  |  02.310$\pm$0.150 |  01.860$\pm$0.170 |  017.0$\pm$9.0
134 |  |  02.080$\pm$0.180 |  01.490$\pm$0.140 |  098.7$\pm$6.2 |  | … | … | …
135 |  |  01.826$\pm$0.074 |  01.119$\pm$0.078 |  166.0$\pm$2.7 |  |  00.813$\pm$0.843 |  00.354$\pm$0.673 |  004.7$\pm$9.8
137 |  G34.43$+$0.2M3 |  03.890$\pm$0.110 |  02.660$\pm$0.110 |  165.1$\pm$1.4 |  |  03.580$\pm$0.120 |  02.190$\pm$0.130 |  166.2$\pm$1.7
138 |  |  01.764$\pm$0.020 |  01.587$\pm$0.020 |  156.0$\pm$1.8 |  |  00.898$\pm$0.045 |  00.483$\pm$0.084 |  172.0$\pm$5.7
139 |  |  02.024$\pm$0.065 |  01.596$\pm$0.054 |  143.8$\pm$2.5 |  |  01.287$\pm$0.087 |  00.594$\pm$0.181 |  149.0$\pm$5.0
140 |  |  01.553$\pm$0.023 |  01.880$\pm$0.061 |  015.4$\pm$8.2 |  |  01.390$\pm$0.100 |  01.040$\pm$0.120 |  021.0$\pm$8.5
141 |  |  03.575$\pm$0.098 |  01.410$\pm$0.024 |  150.5$\pm$2.5 |  | … | … | …
142 |  |  02.217$\pm$0.056 |  02.918$\pm$0.098 |  074.9$\pm$4.1 |  |  03.240$\pm$0.110 |  02.490$\pm$0.120 |  072.7$\pm$4.9
143 |  IRAS 19095$+$0930 |  02.014$\pm$0.039 |  01.445$\pm$0.044 |  115.5$\pm$0.8 |  |  01.250$\pm$0.350 |  00.340$\pm$1.030 |  124.0$\pm$9.0
144 |  |  03.158$\pm$0.079 |  02.577$\pm$0.091 |  087.0$\pm$1.6 |  |  02.724$\pm$0.090 |  02.177$\pm$0.111 |  084.3$\pm$3.7
145 |  |  01.601$\pm$0.011 |  01.370$\pm$0.013 |  094.7$\pm$0.6 |  | … | … | …
146 |  |  01.715$\pm$0.018 |  01.365$\pm$0.021 |  105.1$\pm$0.6 |  | … | … | …
Observed and deconvolved source sizes for K band detections
48a |  IRAS 18228$-$1312 |  02.950$\pm$0.390 |  02.090$\pm$0.220 |  166.0$\pm$9.0 |  | … | … | …
48b |  |  03.070$\pm$0.640 |  02.330$\pm$0.410 |  079.0$\pm$9.0 |  |  03.060$\pm$0.210 |  02.310$\pm$0.160 |  169.0$\pm$9.0
48c |  |  07.619$\pm$0.716 |  01.042$\pm$0.085 |  081.9$\pm$0.9 |  |  02.300$\pm$1.000 |  01.400$\pm$1.000 |  119.0$\pm$9.0
62 |  G19.27$+$0.1M1 |  00.485$\pm$0.019 |  00.393$\pm$0.021 |  134.6$\pm$7.2 |  | … | … | …
63 |  IRAS 18236$-$1205 |  01.670$\pm$0.024 |  00.970$\pm$0.011 |  133.5$\pm$7.9 |  |  01.510$\pm$0.028 |  00.690$\pm$0.017 |  132.5$\pm$9.4
64 |  |  00.868$\pm$0.036 |  00.738$\pm$0.026 |  121.0$\pm$8.9 |  |  00.499$\pm$0.079 |  00.240$\pm$0.168 |  109.0$\pm$9.0
65 |  |  00.799$\pm$0.016 |  00.713$\pm$0.013 |  109.9$\pm$6.5 |  |  00.394$\pm$0.040 |  00.035$\pm$0.136 |  093.2$\pm$9.2
73 |  G23.60$+$0.0M1 |  00.598$\pm$0.087 |  00.365$\pm$0.035 |  054.1$\pm$7.5 |  | … | … | …
74 |  |  01.022$\pm$0.279 |  00.397$\pm$0.061 |  018.9$\pm$5.7 |  |  00.878$\pm$0.356 |  00.226$\pm$0.111 |  015.0$\pm$9.0
83 |  IRAS 18316$-$0602 |  01.069$\pm$0.105 |  00.611$\pm$0.048 |  178.4$\pm$5.4 |  |  00.969$\pm$0.126 |  00.451$\pm$0.096 |  171.2$\pm$9.2
94 |  |  00.946$\pm$0.284 |  00.682$\pm$0.180 |  057.0$\pm$9.0 |  |  00.809$\pm$0.374 |  00.576$\pm$0.511 |  070.0$\pm$9.0
95 |  |  00.455$\pm$0.081 |  00.388$\pm$0.059 |  016.0$\pm$9.0 |  | … | … | …
96 |  G24.08$+$0.0M2 |  00.503$\pm$0.021 |  00.407$\pm$0.023 |  027.4$\pm$7.8 |  | … | … | …
110 |  G24.33$+$0.1M1 |  00.528$\pm$0.018 |  00.350$\pm$0.020 |  033.4$\pm$3.3 |  | … | … | …
136 |  G24.60$+$0.01M2 |  00.685$\pm$0.068 |  00.492$\pm$0.068 |  078.0$\pm$9.0 |  | … | … | …
137 |  G34.43$+$0.2M3 |  01.025$\pm$0.060 |  00.797$\pm$0.070 |  061.0$\pm$9.0 |  |  00.899$\pm$0.085 |  00.715$\pm$0.108 |  123.8$\pm$3.9
143 |  IRAS 19095$+$0930 |  00.572$\pm$0.012 |  00.371$\pm$0.012 |  050.0$\pm$1.9 |  |  00.269$\pm$0.034 |  00.181$\pm$0.058 |  094.0$\pm$9.0
Table 6: continued. Table 7: Intensities, fluxes and source sizes for the sources detected at both frequency bands and imaged using the common uv-range (see Sect. 3.2) |  |  |  | Convolved Image Size |  | Deconvolved Image Size |  |
---|---|---|---|---|---|---|---|---
|  |  Intensityaa$a$Flux density, intensity peak and deconvolved source size for the sources detected at 6 cm in the C band images. The fluxes are corrected by the primary beam response of the antennas, except for sources located outside the primary beam (listed as ‘oC’). Source sizes are calculated as indicated in Appendix B and based on the values reported in Table LABEL:t:sizes. Upper limits in the source size indicate that the source could not be deconvolved (see more details in Table LABEL:t:sizes). |  Fluxaa$a$The intensities and fluxes are corrected by the primary beam response of the antennas, except for source #137 which is located outside the primary beam of the antennas and no correction factor can be determined. | Major Axis | Minor Axis | PA |  | Major Axis | Minor Axis | PA |  $\theta_{\mathrm{beam}}$, PA | rms
ID | Region |  (mJy beam-1) | (mJy) | (″) | (″) | (deg) |  | (″) | (″) | (deg) |  (″$\times$″, °) | (mJy)
Intensities, fluxes and sizes from the C band images with common uv-range |  |
48a |  IRAS 18228$-$1312 |  011.50$\pm$0.61 |  55.32$\pm$3.50 |  04.60$\pm$0.28 |  02.08$\pm$0.10 |  000$\pm$20 |  |  04.24$\pm$0.31 |  01.72$\pm$0.13 |  002$\pm$30 |  1.88$\pm$1.66, $-$14 | 0.400
48b |  |  008.76$\pm$0.99 |  75.41$\pm$9.50 |  04.35$\pm$0.50 |  03.93$\pm$0.44 |  040$\pm$44 |  |  04.15$\pm$0.72 |  03.56$\pm$0.65 |  052$\pm$54 |  1.88$\pm$1.66, $-$14 | 0.400
48c |  |  032.31$\pm$1.70 |  129.52$\pm$8.30 |  03.21$\pm$0.18 |  02.48$\pm$0.12 |  061$\pm$80 |  |  03.00$\pm$0.21 |  01.73$\pm$0.21 |  066$\pm$80 |  1.88$\pm$1.66, $-$14 | 0.400
63 |  IRAS 18236$-$1205 |  000.56$\pm$0.02 |  000.99$\pm$0.06 |  02.24$\pm$0.11 |  01.66$\pm$0.07 |  162$\pm$50 |  |  01.26$\pm$0.17 |  01.13$\pm$0.28 |  071$\pm$63 |  1.99$\pm$1.11, $-$18 | 0.021
64 |  |  000.30$\pm$0.01 |  000.28$\pm$0.02 |  01.97$\pm$0.09 |  00.97$\pm$0.02 |  168$\pm$10 |  | … | … | … |  1.94$\pm$1.08, $-$18 | 0.005
65 |  |  000.56$\pm$0.07 |  000.57$\pm$0.14 |  01.99$\pm$0.04 |  01.07$\pm$0.01 |  163$\pm$10 |  | … | … | … |  1.94$\pm$1.08, $-$18 | 0.004
73 |  G23.60$+$0.0M1 |  000.22$\pm$0.08 |  000.24$\pm$0.15 |  01.82$\pm$0.09 |  01.14$\pm$0.03 |  159$\pm$20 |  |  00.48$\pm$0.28 |  00.30$\pm$0.16 |  013$\pm$41 |  1.77$\pm$1.08, $-$23 | 0.008
74 |  |  000.29$\pm$0.01 |  000.35$\pm$0.03 |  01.94$\pm$0.11 |  01.27$\pm$0.05 |  163$\pm$40 |  |  00.73$\pm$0.28 |  00.38$\pm$0.29 |  025$\pm$40 |  1.85$\pm$1.13, $-$22 | 0.008
83 |  IRAS 18316$-$0602 |  002.21$\pm$0.09 |  003.35$\pm$0.21 |  01.94$\pm$0.09 |  01.36$\pm$0.05 |  173$\pm$40 |  |  01.16$\pm$0.18 |  00.65$\pm$0.29 |  021$\pm$16 |  1.64$\pm$1.05, $-$22 | 0.011
96 |  G24.08$+$0.0M2 |  018.70$\pm$0.30 |  024.97$\pm$0.65 |  01.95$\pm$0.04 |  01.29$\pm$0.02 |  151$\pm$10 |  |  00.98$\pm$0.09 |  00.57$\pm$0.10 |  125$\pm$10 |  1.73$\pm$1.08, $-$21 | 0.624
110 |  G24.33$+$0.1M1 |  000.46$\pm$0.12 |  000.46$\pm$0.21 |  01.55$\pm$0.05 |  01.24$\pm$0.03 |  168$\pm$40 |  | … | … | … |  1.61$\pm$1.19, $-$13 | 0.009
137 |  G34.43$+$0.2M3 |  003.41$\pm$0.23 |  014.40$\pm$1.20 |  03.36$\pm$0.24 |  02.54$\pm$0.17 |  170$\pm$90 |  |  03.04$\pm$0.28 |  02.10$\pm$0.21 |  171$\pm$10 |  1.46$\pm$1.38, $-$59 | 0.242
143 |  IRAS 19095$+$0930 |  030.03$\pm$0.74 |  039.50$\pm$1.60 |  01.88$\pm$0.05 |  01.34$\pm$0.03 |  113$\pm$30 |  |  01.18$\pm$0.01 |  00.26$\pm$0.17 |  121$\pm$50 |  1.49$\pm$1.28, $-$85 | 0.107
Intensities, fluxes and sizes from the K band images with the common uv-range |  |
48a |  IRAS 18228$-$1312 |  001.51$\pm$0.13 |  015.19$\pm$1.51 |  03.02$\pm$0.28 |  01.62$\pm$0.13 |  001$\pm$50 |  |  02.93$\pm$0.29 |  01.47$\pm$0.15 |  001$\pm$50 |  2.50$\pm$1.25, $+$57 | 0.200
48b |  |  001.59$\pm$0.17 |  010.53$\pm$1.30 |  02.09$\pm$0.23 |  01.54$\pm$0.16 |  067$\pm$13 |  |  01.98$\pm$0.25 |  01.36$\pm$0.18 |  067$\pm$13 |  2.50$\pm$1.25, $+$57 | 0.200
48c |  |  006.28$\pm$0.36 |  054.41$\pm$3.51 |  02.81$\pm$0.17 |  01.50$\pm$0.08 |  067$\pm$30 |  |  02.73$\pm$0.18 |  01.31$\pm$0.09 |  068$\pm$30 |  2.50$\pm$1.25, $+$57 | 0.200
63 |  IRAS 18236$-$1205 |  000.21$\pm$0.03 |  000.75$\pm$0.13 |  01.58$\pm$0.23 |  01.13$\pm$0.14 |  115$\pm$15 |  |  01.41$\pm$0.28 |  00.88$\pm$0.21 |  113$\pm$23 |  0.51$\pm$0.32, $-$26 | 0.002
64 |  |  000.17$\pm$0.03 |  000.35$\pm$0.08 |  01.24$\pm$0.24 |  00.81$\pm$0.12 |  127$\pm$13 |  |  01.01$\pm$0.32 |  00.43$\pm$0.28 |  125$\pm$27 |  0.73$\pm$0.67, $-$25 | 0.004
65 |  |  002.03$\pm$0.05 |  002.31$\pm$0.09 |  00.81$\pm$0.02 |  00.70$\pm$0.02 |  110$\pm$60 |  | … | … | … |  0.73$\pm$0.67, $-$25 | 0.002
73 |  G23.60$+$0.0M1 |  000.44$\pm$0.05 |  000.43$\pm$0.09 |  00.89$\pm$0.12 |  00.61$\pm$0.06 |  074$\pm$10 |  | … | … | … |  0.76$\pm$0.71, 0$-$1 | 0.001
74 |  |  000.20$\pm$0.04 |  000.49$\pm$0.13 |  01.01$\pm$0.27 |  00.39$\pm$0.06 |  018$\pm$50 |  |  00.86$\pm$0.34 |  00.21$\pm$0.11 |  015$\pm$17 |  0.54$\pm$0.30, $+$32 | 0.002
83 |  IRAS 18316$-$0602 |  001.55$\pm$0.09 |  003.73$\pm$0.29 |  01.34$\pm$0.08 |  00.96$\pm$0.05 |  003$\pm$60 |  |  01.11$\pm$0.10 |  00.65$\pm$0.08 |  003$\pm$70 |  0.91$\pm$0.84, $-$47 | 1.924
96 |  G24.08$+$0.0M2 |  004.75$\pm$0.32 |  008.26$\pm$0.83 |  01.18$\pm$0.09 |  00.78$\pm$0.05 |  128$\pm$60 |  |  00.94$\pm$0.12 |  00.24$\pm$0.13 |  125$\pm$70 |  0.73$\pm$0.70, 0$+$6 | 0.008
110 |  G24.33$+$0.1M1 |  000.97$\pm$0.31 |  001.20$\pm$0.06 |  00.93$\pm$0.03 |  00.85$\pm$0.03 |  146$\pm$13 |  |  00.45$\pm$0.11 |  00.30$\pm$0.18 |  001$\pm$37 |  0.85$\pm$0.75, $-$56 | 0.002
137 |  G34.43$+$0.2M3 |  000.53$\pm$0.04 |  002.21$\pm$0.20 |  02.33$\pm$0.19 |  01.50$\pm$0.10 |  179$\pm$60 |  |  01.19$\pm$0.14 |  02.14$\pm$0.21 |  180$\pm$70 |  0.95$\pm$0.88, $-$41 | 0.043
143 |  IRAS 19095$+$0930 |  117.66$\pm$1.40 |  130.87$\pm$2.60 |  00.94$\pm$0.01 |  00.90$\pm$0.01 |  117$\pm$10 |  |  00.34$\pm$0.05 |  00.23$\pm$0.09 |  066$\pm$29 |  0.91$\pm$0.84, $-$47 | 1.924
101010 Figure 20:  Close-up views of the C band (color scale image) and K band
(contours) continuum images for the sources listed in Table LABEL:t:catalogue.
Maps for the sources detected in region IRAS 05358$+$3543. Figure 21:  Close-
up views of the C band (color scale image) and K band (contours) continuum
images for the sources listed in Table LABEL:t:catalogue. Maps for the sources
detected in region G189.78$+$0.34. Figure 22:  Close-up views of the C band
(color scale image) and K band (contours) continuum images for the sources
listed in Table LABEL:t:catalogue. Maps for the sources detected in region
G192.58$-$0.04. Figure 23:  Close-up views of the C band (color scale image)
and K band (contours) continuum images for the sources listed in Table
LABEL:t:catalogue. Maps for the sources detected in region G18.18$-$0.30.
Figure 24:  Close-up views of the C band (color scale image) and K band
(contours) continuum images for the sources listed in Table LABEL:t:catalogue.
Maps for the sources detected in region IRAS 18223$-$1243. Figure 25:  Close-
up views of the C band (color scale image) and K band (contours) continuum
images for the sources listed in Table LABEL:t:catalogue. Maps for the sources
detected in region IRAS 18228$-$1312. Figure 26:  Close-up views of the C band
(color scale image) and K band (contours) continuum images for the sources
listed in Table LABEL:t:catalogue. Maps for the sources detected in region
G19.27$+$0.1 M1. Figure 27:  Close-up views of the C band (color scale image)
and K band (contours) continuum images for the sources listed in Table
LABEL:t:catalogue. Maps for the sources detected in region IRAS 18236$-$1205.
Figure 28:  Close-up views of the C band (color scale image) and K band
(contours) continuum images for the sources listed in Table LABEL:t:catalogue.
Maps for the sources detected in region G23.60$+$0.0 M1.

Figure 29:  Close-up views of the C band (color scale image) and K band
(contours) continuum images for the sources listed in Table LABEL:t:catalogue.
The image of sources #94 and #95 correspond to the K band maps. Maps for the
sources detected in region IRAS 18316$-$0602. Figure 30:  Close-up views of
the C band (color scale image) and K band (contours) continuum images for the
sources listed in Table LABEL:t:catalogue. Maps for the sources detected in
region G24.08$+$0.0 M2. Figure 31:  Close-up views of the C band (color scale
image) and K band (contours) continuum images for the sources listed in Table
LABEL:t:catalogue. Maps for the sources detected in region G24.33$+$0.1 M1.
Figure 32:  Close-up views of the C band (color scale image) and K band
(contours) continuum images for the sources listed in Table LABEL:t:catalogue.
Maps for the sources detected in region G24.60$+$0.1 M1.

Figure 33:  Close-up views of the C band (color scale image) and K band
(contours) continuum images for the sources listed in Table LABEL:t:catalogue.
The image of source #136 corresponds to the K band map. Maps for the sources
detected in region G24.60$+$0.1 M2. Figure 34:  Close-up views of the C band
(color scale image) and K band (contours) continuum images for the sources
listed in Table LABEL:t:catalogue. Maps for the sources detected in region
G34.43$+$0.2 M3. Figure 35:  Close-up views of the C band (color scale image)
and K band (contours) continuum images for the sources listed in Table
LABEL:t:catalogue. Maps for the sources detected in region IRAS 19095$+$0930.


# Interior pointwise $C^{1,\alpha}$ estimates for Stokes systems in divergence
form111Research supported by National Natural Science Foundation of China
[grant 12001419 and grant 12071365] and the Fundamental Research Funds for the
Central Universities [grant xzy012022004].

Rong Dong<EMAIL_ADDRESS>Dongsheng Li<EMAIL_ADDRESS>
###### Abstract

Interior pointwise $C^{1,\alpha}$ estimates are established for Stokes systems
in divergence form where no continuity in time variable is assumed for the
coefficients and the given data. The estimates are attained by iteration and
are presented by Campanato’s characterization. The sharpness of the
conclusions can be seen from Serrin’s counterexample.

###### keywords:

Stokes system, Pointwise $C^{1,\alpha}$ estimate, Campanato’s characterization

###### MSC:

76D07 , 35B65

##  1 Introduction

The primary purpose of this paper is to establish interior pointwise
$C^{1,\alpha}$ estimates for Stokes systems with variable coefficients in
divergence form. Precisely, the system concerned here is the following

$\left\\{\begin{array}[]{ll}\vspace{2mm}u_{t}+\mathcal{L}u+\nabla
p=\mbox{div}f~{}~{}\mbox{in}~{}~{}\Omega_{T};\\\
\vspace{2mm}\mbox{div}~{}u=g~{}~{}\mbox{in}~{}~{}\Omega_{T},\end{array}\right.$
(1.1)

where

$\mathcal{L}:=-\mbox{div}\Big{(}A(x,t)\nabla\Big{)}=:-D_{i}\Big{(}a^{ij}(x,t)D_{j}\Big{)},$

$\Omega_{T}=\Omega\times(-T,0]$ is a cylinder with $T>0$ and
$\Omega\subset\mathbb{R}^{n}$ ($n\geq 2$). Throughout this paper, the
summation convention is assumed and the coefficient matrix
$A(x,t)=\big{(}a^{ij}(x,t)\big{)}_{n\times n}$ always satisfies

($\bf H$) Ellipticity: There exists $\lambda\in(0,1]$ such that

$a^{ij}(x,t)\xi_{i}\xi_{j}\geq\lambda|\xi|^{2}~{}~{}\mbox{and}~{}~{}|a^{ij}(x,t)|\leq\lambda^{-1}$

for any $(x,t)\in\Omega_{T}$ and $\xi=(\xi_{i})\in\mathbb{R}^{n}$.

For Stokes systems with constant coefficients, Solonnikov [14] established the
global $C^{2,\alpha}$ (Schauder) estimates by using potential theory. Later,
he further developed the global $C^{2,\alpha}$ estimates to Stokes systems in
non-divergence form with variable coefficients (cf. [15] and the references
therein). In [1], Abe and Giga showed the analyticity of the Stokes semigroup
in $L^{\infty}_{\sigma}(\Omega)$, where the above estimates played an
important part. For the Stokes systems in divergence form (1.1), the
counterpart is the $C^{1,\alpha}$ estimates and this area seems remaining
almost untouched. Here are some related interesting results with constant
coefficients and divergence free velocities. Chang and Kang [4] solved the
system in Hölder spaces by giving a global Hölder estimate of the velocities.
Naumann and Wolf [12] showed that any weak solution is Hölder continuous with
respect to the spatial variables that belong to $\mathbb{R}^{3}$.

Since Stokes systems are parabolic with degeneration, the global regularity
heavily depends on the boundary data (cf. [1] [4] [14] and [15]) and the
interior regularity is very different from the global ones. This paper will
investigate the interior $C^{1,\alpha}$ regularity for (1.1). From Serrin’s
counterexample (cf. [13] or the following Remark 1.3 (ii)), one may find that
the velocity is not Hölder continuous in the time variable even though the
coefficients are constant and the righthand side terms are vanishing. That is,
the regularity in the time direction is weak. Actually, we establish the
interior pointwise $C^{1,\alpha}$ regularity of velocities in the spatial
variables provided $A$ and $f$ are only Hölder continuous in spatial
variables, and $\nabla g$ is bounded. (This much improves the results in
[12].) However an astonished result is that for curls of velocities, the
Hölder continuity holds with respect to both spatial variables and time
variable. (Observe no Hölder continuity in time variable is assumed for the
coefficients and the righthand side terms.) A corresponding result for $L_{p}$
estimates of (non-degenerate) parabolic equations is given by Krylov [10],
where $W^{2}_{p}$ and $W^{1}_{p}$ estimates were presented for equations in
non-divergence form and divergence form respectively as the coefficients are
only VMO in spatial variables. Such results (cf. Lemma 2.4 in the following)
will be used in our proofs. The conclusions in this paper are similar to
directional regularity which one tries to reveal under conditions possessing
regularity only on some directions. Tian and Wang [17] obtained directional
$C^{2,\alpha}$ regularity for elliptic equations, while Dong and Kim [5]
further gave directional $C^{1,\alpha}$ and $C^{2,\alpha}$ estimates for both
elliptic and parabolic equations. Dong, Li and Wang [6] and [7] studied
directional homogenization problems and found different regularity between
homogenization directions and non-homogenization directions.

Pointwise interior $C^{2,\alpha}$ and $C^{1,\alpha}$ estimates for fully
nonlinear elliptic equations were originated by L.Caffarelli in his celebrated
paper [2]. Roughly speaking, under some structure regularity for the fully
nonlinear elliptic operators, the previous estimate states that the solution
will be $C^{2,\alpha}$ continuous at the same point if the nonlinear elliptic
operator and the righthand side term are Hölder continuous at $x_{0}$; and the
latter gives the $C^{1,\alpha}$ continuity of solutions as the oscillation of
the nonlinear elliptic operator is of small BMO and the righthand side term is
of $C^{\alpha-1}$ at the point concerned. We will use the idea in [2] (cf.
also [3]) to show the pointwise $C^{1,\alpha}$ estimate for the velocity of
(1.1) in the space direction. That is, the velocity is approximated at a
$(1+\alpha)$-th order of the scales by a spatial variable first order
polynomial whose coefficients may depend on the time. The approximation is
indeed Campanato’s characterization of $C^{1,\alpha}$ (cf. [8] [9] and [11])
and the polynomial will be obtained by an iteration. In the iteration, the
velocity, its gradient and its curl should be involved simultaneously and
beyond expectation, for the curl of the velocity, the pointwise $C^{\alpha}$
estimate is achieved. We will estimate the curl of velocity by the gradient of
it via parabolic equations, while estimate the gradient by the curl and
velocity itself via elliptic equations.

The following are the main results of this paper.

######  Theorem 1.1.

Suppose $M_{0},M_{1},M_{2}>0$, $0<\alpha<1$, $q>\frac{n}{1-\alpha}$,
Hypothesis $(H)$ holds, and $u\in L_{2}(-T,0;W^{1}_{2}(\Omega))\cap
L_{\infty}(-T,0;L_{2}(\Omega))$ and $p\in L_{1}(\Omega_{T})$ satisfying (1.1)
in weak sense. If $Q_{R}(x_{0},t_{0})\subset\Omega_{T}$ and for any $0<r<R$,

$\big{|}A(x,t)-A(x_{0},t)\big{|}\leq
M_{0}r^{\alpha}~{}~{}\mbox{in}~{}~{}Q_{r}(x_{0},t_{0}),$ (1.2)
$\fint_{Q_{r}(x_{0},t_{0})}|f-\overline{f}_{B_{r}(x_{0})}|^{2}dxdt\leq
M_{1}r^{2\alpha}$ (1.3)

and

$\sup_{t\in(t_{0}-r^{2},t_{0}]}\fint_{B_{r}(x_{0})}|\nabla g|^{q}dx\leq
M_{2},$ (1.4)

then there exist two constants $0<\sigma<1$ and $C>0$ depending only on $n$,
$\lambda$, $\alpha$, $q$ and $M_{0}$, and two functions $a(t)\in
L_{\infty}(t_{0}-\sigma^{2}R^{2},t_{0};\mathbb{R}^{n})$ and $b(t)\in
L_{\infty}(t_{0}-\sigma^{2}R^{2},t_{0};\mathbb{R}^{n^{2}})$ such that

$\displaystyle||a||_{L_{\infty}(t_{0}-\sigma^{2}R^{2},t_{0})}+||b||_{L_{\infty}(t_{0}-\sigma^{2}R^{2},t_{0})}\leq
C{\cal K},$ $\displaystyle\fint_{Q_{r}(x_{0},t_{0})}|\nabla\times
u-\overline{\nabla\times u}_{Q_{r}(x_{0},t_{0})}|^{2}dxdt\leq
Cr^{2\alpha}{\cal K}^{2}$ (1.5)

and

$\displaystyle\sup_{t\in(t_{0}-r^{2},t_{0}]}\fint_{B_{r}(x_{0})}|u-a(t)-b(t)x|^{2}dx\leq$
$\displaystyle\ Cr^{2(1+\alpha)}{\cal K}^{2}$ (1.6)

for any $0<r\leq\sigma R$, where

${\cal
K}=\|u\|_{L_{\infty}(t_{0}-R^{2},t_{0};L_{2}(B_{R}(x_{0})))}+\|\nabla\times
u\|_{L_{2}(Q_{R}(x_{0},t_{0}))}+M^{\frac{1}{2}}_{1}+M^{\frac{1}{q}}_{2}.$

From the above pointwise results, the classical $C^{1,\alpha}$ estimates
follow directly.

######  Corollary 1.2.

Suppose $M_{0}>0$, $0<\alpha<1$, and $A\in C_{x}^{\alpha}(\Omega_{T})$ with
$[A]_{x,\alpha;\Omega_{T}}\leq M_{0}$ and that $(H)$ holds. Let $u\in
L_{2}(-T,0;W^{1}_{2}(\Omega))\cap L_{\infty}(-T,0;L_{2}(\Omega))$, $p\in
L_{1}(\Omega_{T})$, $f\in C_{x}^{\alpha}(\Omega_{T};\mathbb{R}^{n^{2}})$ and
$g\in L_{\infty}(-T,0;W^{1}_{\infty}(\Omega))$ satisfying (1.1) in weak sense.
Then for any $0<T^{\prime}<T$ and any $\Omega^{\prime}\subset\subset\Omega$,
we have $\nabla\times u\in
C^{\alpha,\frac{\alpha}{2}}(\overline{\Omega^{\prime}_{T^{\prime}}})$ and
$\nabla u\in C_{x}^{\alpha}(\overline{\Omega^{\prime}_{T^{\prime}}})$ such
that

$\displaystyle\|\nabla\times
u\|_{C^{\alpha,\frac{\alpha}{2}}(\overline{\Omega^{\prime}_{T^{\prime}}})}+\|\nabla
u\|_{C_{x}^{\alpha}(\overline{\Omega^{\prime}_{T^{\prime}}})}$ (1.7)
$\displaystyle\mbox{}\hskip 14.22636pt\leq
C\Big{(}\|u\|_{L_{\infty}(-T,0;L_{2}(\Omega))}+\|\nabla\times
u\|_{L_{2}(\Omega_{T})}+\|f\|_{C_{x}^{\alpha}(\Omega_{T})}+\|\nabla
g\|_{L_{\infty}(\Omega_{T})}\Big{)}$

where $C$ depends only on $n$, $\lambda$, $\alpha$, $M_{0}$ and
$\min\\{\mbox{dist}(\Omega^{\prime},\partial\Omega),\sqrt{T-T^{\prime}}\\}$.

######  Remark 1.3.

(i) Theorem 1.1 and Corollary 1.2 are effectively $C^{1,\alpha}$ regularity
for weak solutions of Stokes systems in divergence form.

(ii) Let $h$ be a harmonic function in $\Omega$ and $\sigma$ be a function of
$L_{1}(-T,0)$. Then $u(x,t)=\nabla h(x)\int_{-T}^{t}\sigma(s)ds$ and
$p(x,t)=-h(x)\sigma(t)$ is a pair of weak solution of (1.1) with vanishing $f$
and $g$. (This counterexample was given by Serrin [13].) It is easy to see
that $Du=D\nabla h(x)\int_{-T}^{t}\sigma(s)ds$, which may not be Hölder
continuous in variable $t$. That is, our conclusions are sharp.

(iii) Observe that $\nabla\times u$ has Hölder continuity in $t$ although $A$,
$f$ and $g$ are not assumed to be continuous in $t$.

The remainder of the paper will be organized as follows. In Section 2, we give
some preliminary tools. The proof of Theorem 1.1 and Corollary 1.2 will be
presented in Section 3.

Before the end of this section, we introduce some notations. Notations:

$B_{r}(x):=\\{y\in\mathbb{R}^{n}:|x-y|<r\\}$; $B_{r}:=B_{r}(0)$.

$Q_{r}(x,t):=B_{r}(x)\times(t-r^{2},t]$; $Q_{r}:=B_{r}\times(-r^{2},0]$.

$\overline{u}_{Q_{r}(x,t)}:=\fint_{Q_{r}(x,t)}u(y,s)dyds=\frac{1}{|Q_{r}(x,t)|}\int_{Q_{r}(x,t)}u(y,s)dyds$.

$\overline{u}_{B_{r}(x)}(s):=\fint_{B_{r}(x)}u(y,s)dy=\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}u(y,s)dy$.

$D_{i}u:=\frac{\partial u}{\partial x_{i}}$;
$Du:=\big{(}D_{1}u,D_{2}u,...,D_{n}u\big{)}$;
$D^{2}u:=\big{(}D_{i}D_{j}u\big{)}_{n\times n}$.

$\nabla\times u:=\big{(}D_{j}u_{i}-D_{i}u_{j}\big{)}_{n\times n}$.

$\partial_{p}\Omega_{T}:\mbox{parabolic~{}boundary~{}of}~{}\Omega_{T}$.

$[u]_{\alpha,\frac{\alpha}{2};\Omega_{T}}:=\sup_{(x,t),(y,s)\in\Omega_{T},(x,t)\neq(y,s)}\frac{|u(x,t)-u(y,s)|}{|x-y|^{\alpha}+|t-s|^{\frac{\alpha}{2}}}$.

$\|u\|_{C^{\alpha,\frac{\alpha}{2}}(\Omega_{T})}:=[u]_{\alpha,\frac{\alpha}{2};\Omega_{T}}+\|u\|_{L_{\infty}(\Omega_{T})}$.

$C^{\alpha,\frac{\alpha}{2}}(\Omega_{T}):=\\{u:\|u\|_{C^{\alpha,\frac{\alpha}{2}}(\Omega_{T})}<\infty\\}$.

$[u]_{x,\alpha;\Omega_{T}}:=\sup_{(x,t),(y,t)\in\Omega_{T},x\neq
y}\frac{|u(x,t)-u(y,t)|}{|x-y|^{\alpha}}$.

$\|u\|_{C_{x}^{\alpha}(\Omega_{T})}:=[u]_{x,\alpha;\Omega_{T}}+\|u\|_{L_{\infty}(\Omega_{T})}$.

$C^{\alpha}_{x}(\Omega_{T}):=\\{u:\|u\|_{C_{x}^{\alpha}(\Omega_{T})}<\infty\\}$.

##  2 Preliminary

We first introduce the lemma below, which characterizes the Hölder continuity
by Campanato spaces.

######  Lemma 2.1.

Let $T>0$, $0<\alpha<1$ and $\Omega\subset\mathbb{R}^{n}$ be a bounded
Lipschitz domain.

(i) If

$\displaystyle\sup_{(x,t)\in\Omega_{T}}\sup_{r>0}\Big{(}\fint_{Q_{r}(x,t)\cap\Omega_{T}}|u(y,s)-\overline{u}_{Q_{r}(x,t)\cap\Omega_{T}}|^{2}dyds\Big{)}^{\frac{1}{2}}\leq
Ar^{\alpha}$

for some constant $A$, then $u\in
C^{\alpha,\frac{\alpha}{2}}(\overline{\Omega_{T}})$ and

$[u]_{\alpha,\frac{\alpha}{2};\overline{\Omega_{T}}}\leq CA$

for some constant $C$ depending only on $\alpha$, $n$ and $\Omega_{T}$.

(ii) If there exist constants $r_{0},A>0$ and functions $a(x,t)$ and $b(x,t)$
bounded in $\Omega_{T}$ such that

$\displaystyle\sup_{0<r\leq
r_{0}}\Big{(}\fint_{B_{r}(x)\cap\Omega}|u(y,t)-a(x,t)-b(x,t)y|^{2}dy\Big{)}^{\frac{1}{2}}\leq
Ar^{1+\alpha}$

for any $(x,t)\in\Omega_{T}$, then $\nabla u\in
C_{x}^{\alpha}(\overline{\Omega_{T}})$ and

$[\nabla u]_{x,\alpha;\overline{\Omega_{T}}}\leq CA$

for some constant $C$ depending only on $\alpha$, $n$, $T$ and $\Omega$.

We refer to [11] for the proof of Lemma 2.1 (i), which is for anisotropic
spaces, and to [8] for the proof of Lemma 2.1 (ii). The next Lemma is actually
an equivalence between norms of polynomials, which may be well known.

######  Lemma 2.2.

Let $T>0$, $a(t)\in L_{\infty}(-T,0;\mathbb{R})$ and $b(t)\in
L_{\infty}(-T,0;\mathbb{R}^{n})$. If there exist $A$ and $r>0$ such that

$\displaystyle\sup_{t\in(-T,0]}\left(\fint_{B_{r}}|a(t)-b(t)x|^{2}dx\right)^{\frac{1}{2}}\leq
A,$ (2.1)

then

$\displaystyle\sup_{t\in(-T,0]}|a(t)|\leq
CA~{}~{}\mbox{and}~{}~{}\sup_{t\in(-T,0]}|b(t)|\leq CA/r$ (2.2)

for some constant $C$ depending only on $n$.

Proof. Denote $P_{1}$ the space of one order polynomials and suppose $a+bx\in
P_{1}$ with $a\in\mathbb{R}$ and $b,x\in\mathbb{R}^{n}$. Since both
$\max\\{|a|,|b|\\}$ and $\left(\int_{B_{1}}|a+bx|^{2}dx\right)^{\frac{1}{2}}$
are norms on $P_{1}$, we have they are equivalent, that is, there exists a
constant $C$ depending only on $n$ such that

$C^{-1}\left(\int_{B_{1}}|a+bx|^{2}dx\right)^{\frac{1}{2}}\leq\max\\{|a|,|b|\\}\leq
C\left(\int_{B_{1}}|a+bx|^{2}dx\right)^{\frac{1}{2}}.$

It follows that

$C^{-1}\left(\fint_{B_{r}}|a+bx|^{2}dx\right)^{\frac{1}{2}}\leq\max\\{|a|,r|b|\\}\leq
C\left(\fint_{B_{r}}|a+bx|^{2}dx\right)^{\frac{1}{2}}$

for any $r>0$. Then (2.2) is a consequence of (2.1). ∎

The last two lemmas are $L_{p}$ estimates for elliptic and parabolic equations
and we include them here for convenience.

######  Lemma 2.3.

Let $1<p<\infty$ and $\tilde{\Omega}\subset\subset\Omega$.

(i) If $u\in W^{1}_{p}(\Omega;\mathbb{R}^{n})$, then for any
$\zeta\in\mathbb{R}^{n}$ and $\eta\in\mathbb{R}$ independent of $x$,

$\displaystyle\|\nabla u\|_{L_{p}(\tilde{\Omega})}\leq C\Big{(}\|\nabla\times
u-\zeta\|_{L_{p}(\Omega)}+\|\mbox{div}~{}u-\eta\|_{L_{p}(\Omega)}+\|u\|_{L_{1}(\Omega)}\Big{)},$

where $C$ depends only on $n$, $p$, $\tilde{\Omega}$ and $\Omega$.

(ii) If $u\in W^{2}_{p}(\Omega;\mathbb{R}^{n})$, then

$\displaystyle\|D^{2}u\|_{L_{p}(\tilde{\Omega})}\leq
C\Big{(}\|\mbox{div}~{}(\nabla\times
u)\|_{L_{p}(\Omega)}+\|\nabla(\mbox{div}~{}u)\|_{L_{p}(\Omega)}+\|u\|_{L_{1}(\Omega)}\Big{)},$

where $C$ depends only on $n$, $p$, $\tilde{\Omega}$ and $\Omega$.

Proof. Recall $\nabla\times u=D_{j}u_{i}-D_{i}u_{j}$ and we have the following
identity

$\displaystyle\begin{aligned} \Delta u=\mbox{div}~{}\nabla\times
u+\nabla~{}\mbox{div}~{}u=\mbox{div}~{}(\nabla\times
u-\xi)+\nabla~{}(\mbox{div}~{}u-\eta)~{}~{}\mbox{in}~{}~{}\Omega\end{aligned}$

for any $\zeta\in\mathbb{R}^{n}$ and $\eta\in\mathbb{R}$ independent of $x$.
Then by $W^{1}_{p}$ and $W^{2}_{p}$ estimates for elliptic equations, we
obtain (i) and (ii) immediately. ∎

######  Lemma 2.4.

Let $2\leq p<\infty$, $0<T^{\prime}<T$ and
$\tilde{\Omega}\subset\subset\Omega\subset\mathbb{R}^{n}$ be bounded domains.
Suppose $(a^{ij}(t))_{n\times n}$ (not depending on $x$) satisfies $(H)$.
Consider the equation:

$u_{t}-a^{ij}(t)D_{ij}u=g~{}~{}\mbox{in}~{}~{}\Omega_{T}.$ (2.3)

(i) If $g=0$ and $u$ is a weak solution of (2.3), then $u,Du,u_{t}\in
L_{\infty,loc}(\Omega_{T})$ and

$\displaystyle\|u\|_{L_{\infty}(\tilde{\Omega}_{T^{\prime}})}+\|Du\|_{L_{\infty}(\tilde{\Omega}_{T^{\prime}})}+\|u_{t}\|_{L_{\infty}(\tilde{\Omega}_{T^{\prime}})}\leq
C\|u\|_{L_{p}(\Omega_{T})},$

where $C$ depends only on $n$, $\lambda$, $p$, $T^{\prime}$, $T$,
$\tilde{\Omega}$ and $\Omega$.

(ii) Assume that $\Omega$ is smooth. If $g=D_{ij}f_{ij}$ for some
$f=(f_{ij})\in L_{2}(\Omega_{T};\mathbb{R}^{n^{2}})$ and $u\in
L_{2}(\Omega_{T})$ is the generalized solution of (2.3) with
$u=0~{}\mbox{on}~{}\partial_{p}\Omega_{T}$ (See the following Remark 2.5),
then

$\displaystyle\|u\|_{L_{2}({\Omega}_{T})}\leq C\|f\|_{L_{2}(\Omega_{T})},$
(2.4)

where $C$ depends only on $n$, $\lambda$ and $\Omega_{T}$.

Proof. Let $\Omega^{*}\subset\Omega$ with
$dist(\Omega^{*},\partial\Omega)>\varepsilon_{0}>0$ and
$\tilde{\Omega}\subset\subset\Omega^{*}$. For $0<\varepsilon<\varepsilon_{0}$,
denote $\rho_{\varepsilon}$ the scaled mollifier in $\mathbb{R}^{n}$. Then in
view of $(a^{ij}(t))$ not depending on $x$,
$u_{\varepsilon}=\rho_{\varepsilon}\ast u$ satisfies

$(u_{\varepsilon})_{t}-a^{ij}(t)D_{ij}u_{\varepsilon}=0~{}~{}\mbox{in}~{}~{}\Omega^{*}_{T}.$
(2.5)

Differentiating the equation (2.5) suitable times with respect to $x$, and by
the interior $L_{p}$ estimates for parabolic equations in [10] and embedding
inequalities, we have

$\displaystyle\|u_{\varepsilon}\|_{L_{\infty}(\tilde{\Omega}_{T^{\prime}})}+\|u_{\varepsilon
t}\|_{L_{\infty}(\tilde{\Omega}_{T^{\prime}})}+\|Du_{\varepsilon}\|_{L_{\infty}(\tilde{\Omega}_{T^{\prime}})}\leq
C\|u_{\varepsilon}\|_{L_{p}(\Omega^{*}_{T})},$

where $C$ depends only on $n$, $\lambda$, $p$, $\tilde{\Omega}_{T}$ and
$\Omega^{*}_{T}$. Then (i) is obtained by letting $\varepsilon\rightarrow 0$.

Now we turn to the proof of (ii). Suppose $\\{f_{k}\\}\subset
W^{2,1}_{2}(\Omega_{T})$ such that $f_{k}\rightarrow f$ in $L_{2}$. Then (2.3)
has unique solution sequence $u_{k}\in W^{2,1}_{2}(\Omega_{T})$ with
$u_{k}=0~{}\mbox{on}~{}\partial_{p}\Omega_{T}$. By $L_{2}$ estimates for
parabolic equation, (2.4) holds for $u_{k}$. Let $k\rightarrow\infty$ and we
have (2.4). ∎

######  Remark 2.5.

In the proof of (ii), it is easy to see the limit $u$ does not depend on the
choice of $\\{f_{k}\\}$. We call this $u$ the generalized solution of (2.3)
with $u=0~{}\mbox{on}~{}\partial_{p}\Omega_{T}$.

##  3 Proofs of main results

To prove Theorem 1.1, we need only to show:

######  Theorem 3.1.

Let $T>0$, $0<\alpha<1$ and $q>\frac{n}{1-\alpha}$. Suppose
$Q_{1}\subset\Omega_{T}$, $(H)$ holds, and $u\in
L_{2}(-T,0;W^{1}_{2}(\Omega))\cap L_{\infty}(-T,0;L_{2}(\Omega))$ and $p\in
L_{1}(\Omega_{T})$ satisfying (1.1) in weak sense. There exist constants
$0<r_{0}<1$, $0<\varepsilon_{0}<1$ and $C>0$ depending only on $n$, $\lambda$,
$\alpha$ and $q$ such that if for any $0<r\leq 1$,

$\big{|}A(x,t)-A(0,t)\big{|}\leq\varepsilon_{0}r^{\alpha}~{}~{}\mbox{in}~{}~{}Q_{r},$
(3.1) $\sup_{t\in(-1,0]}\fint_{B_{1}}|u|^{2}dx\leq
1,~{}~{}\fint_{Q_{1}}|\nabla\times u|^{2}dxdt\leq 1,$ (3.2)
$\displaystyle\fint_{Q_{r}}|f-\overline{f}_{B_{r}}|^{2}dxdt\leq\varepsilon^{2}_{0}r^{2\alpha}~{}~{}~{}\mbox{and}$
(3.3) $\displaystyle\sup_{t\in(-r^{2},0]}\fint_{B_{r}}|\nabla g|^{q}dx\leq 1,$
(3.4)

then there are $a(t)\in L_{\infty}(-r^{2}_{0},0;\mathbb{R}^{n})$ and $b(t)\in
L_{\infty}(-r^{2}_{0},0;\mathbb{R}^{n^{2}})$ satisfying

$\sup_{t\in(-r_{0}^{2},0]}|a(t)|~{}~{}\mbox{and}~{}~{}\sup_{t\in(-r_{0}^{2},0]}|b(t)|\leq
C,$ (3.5) $\fint_{Q_{r}}|\nabla\times u-\overline{(\nabla\times
u)}_{Q_{r}}|^{2}dxdt\leq Cr^{2\alpha}~{}~{}\mbox{and}$ (3.6)
$\displaystyle\sup_{t\in(-r^{2},0]}\fint_{B_{r}}|u-a(t)-b(t)x|^{2}dx\leq
Cr^{2(1+\alpha)}$ (3.7)

for any $0<r\leq r_{0}$.

Actually, Theorem 1.1 follows from Theorem 3.1 by the following way of
normalization. After suitable choice of coordinates, we may suppose
$(x_{0},t_{0})=(0,0)$.

Choose $0<\tau<R$ such that $\tau^{\alpha}M_{0}\leq\varepsilon_{0}$. For
$(x,t)\in Q_{1}$ and $0<r\leq 1$, set

$\tilde{A}(x,t)=A(\tau x,\tau^{2}t),~{}~{}\tilde{u}(x,t)=\frac{u(\tau
x,\tau^{2}t)}{\tau J},~{}~{}\tilde{p}(x,t)=\frac{p(\tau x,\tau^{2}t)}{J},$
$\tilde{f}(x,t)=\frac{f(\tau
x,\tau^{2}t)}{J}~{}~{}\mbox{and}~{}~{}\tilde{g}(x,t)=\frac{g(\tau
x,\tau^{2}t)}{J},$

where

$\displaystyle\begin{aligned} J=&\
\frac{1}{\tau|B_{\tau}|^{\frac{1}{2}}}\|u\|_{L_{\infty}(-T,0;L_{2}(\Omega))}+\frac{1}{|Q_{\tau}|^{\frac{1}{2}}}\|\nabla\times
u\|_{L_{2}(\Omega_{T})}+\frac{1}{\varepsilon_{0}}M^{\frac{1}{2}}_{1}+M^{\frac{1}{q}}_{2}.\end{aligned}$

Then $\tilde{u}$, $\tilde{p}$, $\tilde{f}$ and $\tilde{g}$ satisfy (1.1),

$\big{|}\tilde{A}(x,t)-\tilde{A}(0,t)\big{|}=\big{|}A(\tau
x,\tau^{2}t)-A(0,\tau^{2}t)\big{|}\leq
M_{0}\tau^{\alpha}r^{\alpha}\leq\varepsilon_{0}r^{\alpha}~{}~{}\mbox{in}~{}~{}Q_{r},$
$\sup_{t\in(-1,0]}\fint_{B_{1}}|\tilde{u}|^{2}dx=\frac{1}{\tau^{2}J^{2}}\sup_{t\in(-\tau^{2},0]}\fint_{B_{\tau}}|u|^{2}dx\leq
1,$
$\fint_{Q_{1}}|\nabla\times\tilde{u}|^{2}dxdt=\frac{1}{J^{2}}\fint_{Q_{\tau}}|\nabla\times
u|^{2}dxdt\leq 1,$
$\displaystyle\fint_{Q_{r}}|\tilde{f}-\overline{\tilde{f}}_{B_{r}}|^{2}dxdt=\frac{1}{J^{2}}\fint_{Q_{\tau
r}}|f-\overline{f}_{B_{\tau r}}|^{2}dxdt\leq\varepsilon^{2}_{0}r^{2\alpha}$

and

$\displaystyle\sup_{t\in(-r^{2},0]}\fint_{B_{r}}|\nabla\tilde{g}|^{q}dx=\frac{\tau^{q}}{J^{q}}\sup_{t\in(-\tau^{2}r^{2},0]}\fint_{B_{\tau
r}}|\nabla g|^{q}dx\leq 1$

for any $0<r\leq 1$. By Theorem 3.1, there exist $\tilde{a}(t)\in
L_{\infty}(-r^{2}_{0},0;\mathbb{R}^{n})$ and $\tilde{b}(t)\in
L_{\infty}(-r^{2}_{0},0;\mathbb{R}^{n^{2}})$ such that

$\fint_{Q_{r}}|\nabla\times\tilde{u}-\overline{(\nabla\times\tilde{u})}_{Q_{r}}|^{2}dxdt\leq
Cr^{2\alpha}$

and

$\displaystyle\sup_{t\in(-r^{2},0]}\fint_{B_{r}}|\tilde{u}-\tilde{a}(t)-\tilde{b}(t)x|^{2}dx\leq
Cr^{2(1+\alpha)}$

for any $0<r<1$, where $C$ depends only on $n$, $\lambda$, $\alpha$ and $q$.
After rescaling back, we get (1.5) and (1.6) with $a(t)=\tau J\tilde{a}(t)$
and $b(t)=J\tilde{b}(t)$.

We will prove Theorem 3.1 by an iteration staring with the following lemma.

######  Lemma 3.2.

Under the hypotheses of Theorem 3.1, there exist constants
$0<r_{0}\leq\frac{1}{6}$, $0<\hat{\varepsilon}_{0}<1$ and $C_{0}>0$ depending
only on $n$, $\lambda$, $\alpha$ and $q$ such that if

$\big{|}A(x,t)-A(0,t)\big{|}\leq\hat{\varepsilon}_{0}~{}~{}\mbox{in}~{}~{}Q_{1},$
(3.8) $\sup_{t\in(-1,0]}\fint_{B_{1}}|u|^{2}dx\leq
1,~{}~{}\fint_{Q_{1}}|\nabla\times u|^{2}dxdt\leq 1,$ (3.9)
$\fint_{Q_{1}}|f|^{2}dxdt\leq\hat{\varepsilon}^{2}_{0}~{}~{}\mbox{and}~{}~{}\sup_{t\in(-1,0]}\fint_{B_{1}}|\nabla
g|^{q}dx\leq 1,$ (3.10)

then there is $b_{0}(t)\in L_{\infty}(-r^{2}_{0},0;\mathbb{R}^{n^{2}})$
satisfying

$\sup_{t\in(-r_{0}^{2},0]}|b_{0}(t)|\leq C_{0},$ (3.11)
$\fint_{Q_{r_{0}}}|\nabla\times u-\overline{(\nabla\times
u)}_{Q_{r_{0}}}|^{2}dxdt\leq r^{2\alpha}_{0}$ (3.12)

and

$\displaystyle\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|u-\overline{u}_{B_{r_{0}}}(t)-b_{0}(t)x|^{2}dx\leq
r^{2+2\alpha}_{0}.$ (3.13)

Proof. Suppose (3.8)-(3.10) hold. We divide the proof of (3.11)-(3.13) into 7
steps and the constants $r_{0}$, $\hat{\varepsilon}_{0}$ and $C_{0}$ will be
determined in Step 7.

_Step 1. Estimates of $\nabla u$._ Using Lemma 2.3 (i) to $u$ in $B_{1}$ with
$p=2$, $\zeta=0$ and $\eta=\overline{g}_{B_{1}}(t)$, we have, by
$\mbox{div}~{}u=g$,

$\displaystyle\begin{aligned} \int_{B_{\frac{3}{4}}}|\nabla u|^{2}dx\leq&\
C\Big{(}\int_{B_{1}}|\nabla\times
u|^{2}dx+\int_{B_{1}}|g-\overline{g}_{B_{1}}(t)|^{2}dx+\int_{B_{1}}|u|^{2}dx\Big{)}\end{aligned}$

for $t\in(-\frac{9}{16},0]$, where $C$ depends only on $n$. Integrating with
respect to $t$ and by Poincaré’s inequality,

$\displaystyle\begin{aligned} \int_{Q_{\frac{3}{4}}}|\nabla u|^{2}dxdt\leq&\
C\Big{(}\int_{Q_{1}}|\nabla\times u|^{2}dxdt+\int_{Q_{1}}|\nabla
g(t)|^{2}dxdt+\int_{Q_{1}}|u|^{2}dxdt\Big{)}.\end{aligned}$

In view of (3.9) and (3.10), we obtain

$\displaystyle\begin{aligned} \int_{Q_{\frac{3}{4}}}|\nabla u|^{2}dxdt\leq
C\end{aligned}$

for some $C$ depending only on $n$.

From (1.1), one has

$\displaystyle\left\\{\begin{array}[]{ll}\vspace{2mm}u_{t}-\mbox{div}\big{(}A(0,t)\nabla
u\big{)}+\nabla p=\mbox{div}\Big{(}\big{(}A(x,t)-A(0,t)\big{)}\nabla
u+f\Big{)}~{}~{}\mbox{in}~{}~{}Q_{1},\\\
\vspace{2mm}\mbox{div}~{}u=g~{}~{}\mbox{in}~{}~{}Q_{1}.\end{array}\right.$
(3.16)

Let $h(x,t)=\big{(}A(x,t)-A(0,t)\big{)}\nabla u$. From (3.8), it follows that

$\displaystyle\begin{aligned}
\int_{Q_{\frac{3}{4}}}|h|^{2}dxdt=\int_{Q_{\frac{3}{4}}}|\big{(}A(x,t)-A(0,t)\big{)}\nabla
u|^{2}dxdt\leq C^{*}\hat{\varepsilon}_{0}^{2},\end{aligned}$ (3.17)

where $C^{*}$ depends only on $n$.

_Step 2. Proof of ( 3.12)._ Taking $\nabla\times$ on both sides of the first
equation in (3.16) and by $\nabla\times\nabla p=0$, we obtain

$(\nabla\times u)_{t}-\mbox{div}\big{(}A(0,t)\nabla(\nabla\times
u)\big{)}=\nabla\times\mbox{div}\big{(}h+f\big{)}~{}~{}\mbox{in}~{}~{}Q_{1}.$

Decompose $\nabla\times u$ into $w+v$ with $w$ solving

$\left\\{\begin{array}[]{ll}w_{t}-\mbox{div}\big{(}A(0,t)\nabla
w\big{)}=\nabla\times\mbox{div}~{}\big{(}h+f\big{)}~{}~{}\mbox{in}~{}~{}Q_{\frac{3}{4}},\\\
\vspace{2mm}w=0~{}~{}\mbox{on}~{}~{}\partial_{p}Q_{\frac{3}{4}}.\end{array}\right.$
(3.18)

Then it follows that

$\left\\{\begin{array}[]{ll}v_{t}-\mbox{div}\big{(}A(0,t)\nabla
v\big{)}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{3}{4}},\\\ \vspace{2mm}v=\nabla\times
u~{}~{}\mbox{on}~{}~{}\partial_{p}Q_{\frac{3}{4}}.\end{array}\right.$ (3.19)

Applying Lemma 2.4 (ii) to (3.18) and in view of (3.10) and (3.17), we have

$\displaystyle\begin{aligned} \int_{Q_{\frac{1}{2}}}|w|^{2}dxdt\leq
C\int_{Q_{\frac{3}{4}}}|h+f|^{2}dxdt\leq
C\hat{\varepsilon}_{0}^{2}\end{aligned}$ (3.20)

and then

$\displaystyle\begin{aligned}
\fint_{Q_{r_{0}}}|w-\overline{w}_{Q_{r_{0}}}|^{2}dxdt\leq\frac{C}{r^{n+2}_{0}}\int_{Q_{\frac{1}{2}}}|w|^{2}dxdt\leq
C_{1}\frac{\hat{\varepsilon}_{0}^{2}}{r^{n+2}_{0}}\end{aligned}$ (3.21)

for some $C_{1}$ depending only on $n$ and $\lambda$.

Apply Lemma 2.4 (i) to (3.19) and then

$\displaystyle\begin{aligned}
\fint_{Q_{r_{0}}}|v-\overline{v}_{Q_{r_{0}}}|^{2}dxdt\leq&\
\|v-\overline{v}_{Q_{r_{0}}}\|^{2}_{L_{\infty}(Q_{r_{0}})}\\\ \leq&\
Cr^{2}_{0}\Big{(}\|Dv\|_{L_{\infty}(Q_{\frac{1}{4}})}+\|v_{t}\|_{L_{\infty}(Q_{\frac{1}{4}})}\Big{)}^{2}\leq
Cr^{2}_{0}\|v\|^{2}_{L_{2}(Q_{\frac{1}{2}})},\end{aligned}$

where $C$ depends only on $n$ and $\lambda$. From (3.9), (3.20) and
$v=\nabla\times u-w$, it follows that

$\displaystyle\fint_{Q_{r_{0}}}|v-\overline{v}_{Q_{r_{0}}}|^{2}dxdt\leq$
$\displaystyle\ Cr^{2}_{0}\Big{(}\|\nabla\times
u\|^{2}_{L_{2}(Q_{\frac{1}{2}})}+\|w\|^{2}_{L_{2}(Q_{\frac{1}{2}})}\Big{)}\leq
C_{2}r^{2}_{0},$ (3.22)

where $C_{2}$ depends only on $n$ and $\lambda$.

Combining with (3.21), we get

$\displaystyle\fint_{Q_{r_{0}}}|\nabla\times u-\overline{(\nabla\times
u)}_{Q_{r_{0}}}|^{2}dxdt$ (3.23) $\displaystyle\leq$ $\displaystyle\
2\fint_{Q_{r_{0}}}|w-\overline{w}_{Q_{r_{0}}}|^{2}dxdt+2\fint_{Q_{r_{0}}}|v-\overline{v}_{Q_{r_{0}}}|^{2}dxdt$
$\displaystyle\leq$ $\displaystyle\
2C_{1}\frac{\hat{\varepsilon}_{0}^{2}}{r^{n+2}_{0}}+2C_{2}r^{2}_{0}.$

_Step 3. Decomposition._ We decompose $u$ and $p$, satisfying (3.16), by the
following way. Let $(\hat{w},p_{\hat{w}})$ be the solution of :

$\displaystyle\left\\{\begin{array}[]{ll}\vspace{2mm}\hat{w}_{t}-\mbox{div}\big{(}A(0,t)\nabla\hat{w}\big{)}+\nabla
p_{\hat{w}}=\mbox{div}\big{(}h+f\big{)}~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\mbox{div}~{}\hat{w}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\hat{w}=0~{}~{}\mbox{on}~{}~{}\partial_{p}Q_{\frac{1}{2}}.\end{array}\right.$
(3.27)

Set $\hat{v}=u-\hat{w}$ and $p_{\hat{v}}=p-p_{\hat{w}}$. It follows that

$\displaystyle\left\\{\begin{array}[]{ll}\vspace{2mm}\hat{v}_{t}-\mbox{div}\big{(}A(0,t)\nabla\hat{v}\big{)}+\nabla
p_{\hat{v}}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\mbox{div}~{}\hat{v}=g~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\hat{v}=u~{}~{}\mbox{on}~{}~{}\partial_{p}Q_{\frac{1}{2}}.\end{array}\right.$
(3.31)

_Step 4. Estimates of $\hat{w}$._ Multiplying the first equation in (3.27) by
$\hat{w}$ and then taking integral with respect to $x$, we deduce, by using
the other two equations in (3.27),

$\displaystyle\begin{aligned}
\frac{d}{dt}\int_{B_{\frac{1}{2}}}|\hat{w}|^{2}dx+\int_{B_{\frac{1}{2}}}|\nabla\hat{w}|^{2}dx\leq
C\int_{B_{\frac{1}{2}}}|h+f|^{2}dx\end{aligned}$

for $t\in(-\frac{1}{4},0]$. Next taking integral with respect to $t$,

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{4},0]}\int_{B_{\frac{1}{2}}}|\hat{w}|^{2}dx+\int_{Q_{\frac{1}{2}}}|\nabla\hat{w}|^{2}dxdt\leq
C\int_{Q_{\frac{1}{2}}}|h+f|^{2}dxdt\leq
C\hat{\varepsilon}_{0}^{2},\end{aligned}$ (3.32)

where (3.10) and (3.17) are used to derive the last inequality and $C$ depends
only on $n$ and $\lambda$. This implies

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{4},0]}\fint_{B_{r_{0}}}|\hat{w}-\overline{\hat{w}}_{B_{r_{0}}}(t)|^{2}dx\leq\frac{C}{r^{n}_{0}}\sup_{t\in(-\frac{1}{4},0]}\int_{B_{\frac{1}{2}}}|\hat{w}|^{2}dx\leq
C_{3}\frac{\hat{\varepsilon}_{0}^{2}}{r^{n}_{0}},\end{aligned}$ (3.33)

where $C_{3}$ depends only on $n$ and $\lambda$.

_Step 5. Estimates of $\hat{v}$._ It follows from $\hat{v}=u-\hat{w}$, (3.9)
and (3.32) that

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{4},0]}\|\hat{v}\|_{L_{2}(B_{\frac{1}{2}})}\leq\sup_{t\in(-\frac{1}{4},0]}\|u\|_{L_{2}(B_{\frac{1}{2}})}+\sup_{t\in(-\frac{1}{4},0]}\|\hat{w}\|_{L_{2}(B_{\frac{1}{2}})}\leq
C,\end{aligned}$ (3.34)

where $C$ depends only on $n$ and $\lambda$.

Take $\nabla\times$ on both sides of the first equation in (3.31) and in view
of $\nabla\times\nabla p_{\hat{v}}=0$ , one has

$\displaystyle\begin{array}[]{ll}\vspace{2mm}\big{(}\nabla\times\hat{v}\big{)}_{t}-\mbox{div}\Big{(}A(0,t)\nabla\big{(}\nabla\times\hat{v}\big{)}\Big{)}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{3}}.\end{array}$

From Lemma 2.4 (i) and $\nabla\times\hat{v}=\nabla\times
u-\nabla\times\hat{w}$, we deduce

$\displaystyle\begin{aligned}
&\sup_{t\in(-\frac{1}{16},0]}\|\nabla\times\hat{v}\|_{L_{2}(B_{\frac{1}{4}})}+\sup_{t\in(-\frac{1}{16},0]}\|D(\nabla\times\hat{v})\|_{L_{q}(B_{\frac{1}{4}})}\\\
&\mbox{}\hskip 22.76228pt\leq
C\left(\|\nabla\times\hat{v}\|_{L_{\infty}(Q_{\frac{1}{4}})}+\|D(\nabla\times\hat{v})\|_{L_{\infty}(Q_{\frac{1}{4}})}\right)&\\\
&\mbox{}\hskip 56.9055pt\leq
C\|\nabla\times\hat{v}\|_{L_{2}(Q_{\frac{1}{2}})}\leq C\Big{(}\|\nabla\times
u\|_{L_{2}(Q_{\frac{1}{2}})}+\|\nabla\hat{w}\|_{L_{2}(Q_{\frac{1}{2}})}\Big{)}.\end{aligned}$

From (3.9) and (3.32), it follows that

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{16},0]}\|\nabla\times\hat{v}\|_{L_{2}(B_{\frac{1}{4}})}+\sup_{t\in(-\frac{1}{16},0]}\|D(\nabla\times\hat{v})\|_{L_{q}(B_{\frac{1}{4}})}\leq
C\end{aligned}$ (3.36)

for some $C$ depending only on $n$ and $\lambda$.

Apply Lemma 2.3 (i) to $\hat{v}$ with $p=2$, $\zeta=0$ and
$\eta=\overline{g}_{B_{1}}(t)$. By $\mbox{div}~{}\hat{v}=g$, we have

$\displaystyle\begin{aligned}
&\sup_{t\in(-\frac{1}{36},0]}\|\nabla\hat{v}\|_{L_{2}(B_{\frac{1}{6}})}\leq
C\sup_{t\in(-\frac{1}{16},0]}\Big{(}\|\nabla\times\hat{v}\|_{L_{2}(B_{\frac{1}{4}})}+\|g-\overline{g}_{B_{1}}(t)\|_{L_{2}(B_{\frac{1}{4}})}+\|\hat{v}\|_{L_{2}(B_{\frac{1}{4}})}\Big{)}\\\
&\mbox{}\hskip 22.76228pt\leq
C\sup_{t\in(-\frac{1}{16},0]}\Big{(}\|\nabla\times\hat{v}\|_{L_{2}(B_{\frac{1}{4}})}+\|\nabla
g(t)\|_{L_{2}(B_{\frac{1}{4}})}+\|\hat{v}\|_{L_{2}(B_{\frac{1}{4}})}\Big{)}\leq
C,\end{aligned}$ (3.37)

where the second inequality is derived from Poincaré’s inequality and the last
inequality is derived from (3.10), (3.34) and (3.36).

Applying Lemma 2.3 (ii) to $\hat{v}$ with $p=q$ and by (3.10), (3.34) and
(3.36), we have

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{36},0]}\|D^{2}\hat{v}\|_{L_{q}(B_{\frac{1}{6}})}\leq&\
C\sup_{t\in(-\frac{1}{16},0]}\Big{(}\|D(\nabla\times\hat{v})\|_{L_{q}(B_{\frac{1}{4}})}+\|\nabla
g\|_{L_{q}(B_{\frac{1}{4}})}+\|\hat{v}\|_{L_{2}(B_{\frac{1}{4}})}\Big{)}\leq
C,\end{aligned}$

which leads to

$\displaystyle\begin{aligned}
\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|\hat{v}-\overline{\hat{v}}_{B_{r_{0}}}(t)-\overline{\nabla\hat{v}}_{B_{r_{0}}}(t)x|^{2}dx$\mbox{}\hfil\lx@intercol\hskip
45.5244pt\\\ \leq&\
Cr^{4}_{0}\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|D^{2}\hat{v}|^{2}dx\leq
Cr^{4}_{0}\sup_{t\in(-r_{0}^{2},0]}\Big{(}\fint_{B_{r_{0}}}|D^{2}\hat{v}|^{q}dx\Big{)}^{\frac{2}{q}}\\\
\leq&\
Cr^{4-\frac{2n}{q}}_{0}\sup_{t\in(-r_{0}^{2},0]}\|D^{2}\hat{v}\|^{2}_{L_{q}(B_{\frac{1}{6}})}\leq
C_{4}r^{4-\frac{2n}{q}}_{0},\end{aligned}$ (3.38)

where $C_{4}$ depends only on $n$, $\lambda$ and $q$.

_Step 6. Proof of ( 3.11) and (3.13)._ Set
$b_{0}(t)=\overline{\nabla\hat{v}}_{B_{r_{0}}}(t)$. It follows from (3.37)
that

$\displaystyle\sup_{t\in(-r_{0}^{2},0]}|b_{0}(t)|\leq$ $\displaystyle\
\sup_{t\in(-r_{0}^{2},0]}\Big{(}\fint_{B_{r_{0}}}|\nabla\hat{v}|^{2}dx\Big{)}^{\frac{1}{2}}$
(3.39) $\displaystyle\leq$ $\displaystyle\
\frac{C}{r^{\frac{n}{2}}_{0}}\sup_{t\in(-r_{0}^{2},0]}\Big{(}\int_{B_{\frac{1}{6}}}|\nabla\hat{v}|^{2}dx\Big{)}^{\frac{1}{2}}\leq
C_{5}r^{-\frac{n}{2}}_{0}.$

where $C_{5}$ depends only on $n$ and $\lambda$.

Combining (3.33) and (3.38), we get

$\displaystyle\begin{aligned}
&\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|u-\overline{u}_{B_{r_{0}}}(t)-b_{0}(t)x|^{2}dx~{}~{}~{}~{}~{}(\mbox{by}~{}u=\hat{v}+\hat{w})\\\
&\leq
2\sup_{t\in(-r_{0}^{2},0]}\left(\fint_{B_{r_{0}}}|\hat{v}-\overline{\hat{v}}_{B_{r_{0}}}(t)-\overline{\nabla\hat{v}}_{B_{r_{0}}}(t)x|^{2}dx+\fint_{B_{r_{0}}}|\hat{w}-\overline{\hat{w}}_{B_{r_{0}}}(t)|^{2}dx\right)\\\
&\leq
2C_{4}r^{4-\frac{2n}{q}}_{0}+2C_{3}\frac{\hat{\varepsilon}_{0}^{2}}{r^{n}_{0}}.\end{aligned}$
(3.40)

_Step 7. Determination of the constants._ Let $C_{i}$ ($i=1\sim 5$) be given
by (3.19), (3.20), (3.25), (3.29) and (3.30) respectively. Recall $0<\alpha<1$
and $q>\frac{n}{1-\alpha}$. We first set $0<r_{0}\leq\frac{1}{6}$ small enough
such that

$\displaystyle
2C_{2}r^{2}_{0}\leq\frac{1}{2}r_{0}^{2\alpha}~{}~{}\mbox{and}~{}~{}2C_{4}r^{4-\frac{2n}{q}}_{0}\leq\frac{1}{2}r_{0}^{2+2\alpha}.$
(3.41)

Next, we set $0<\hat{\varepsilon}_{0}<1$ small enough such that

$\displaystyle
2C_{1}\hat{\varepsilon}_{0}^{2}r^{-n-2}_{0}\leq\frac{1}{2}r_{0}^{2\alpha}~{}~{}\mbox{and}~{}~{}2C_{3}\hat{\varepsilon}_{0}^{2}{r^{-n}_{0}}\leq\frac{1}{2}r_{0}^{2+2\alpha}.$
(3.42)

Then (3.12) and (3.13) follow from (3.23) and (3.40) respectively. Finally,
set $C_{0}=C_{5}r^{-\frac{n}{2}}_{0}$ and then (3.11) follows from (3.39). ∎

######  Lemma 3.3.

Suppose the hypotheses of Theorem 3.1 hold. Let $r_{0}$,
$\hat{\varepsilon}_{0}$ and $C_{0}$ be given by Lemma 3.2 and

$\hat{C}_{0}=C_{0}+\bar{C}_{0}r^{-\frac{n}{2}}_{0}\sum_{k=1}^{\infty}r^{k\alpha}_{0}$
(3.43)

with some constant $\bar{C}_{0}>0$ depending only on $n$ and $\lambda$. There
exists a constant $0<\varepsilon_{0}\leq\hat{\varepsilon}_{0}$ depending only
on $n$, $\lambda$, $\alpha$ and $q$ such that if (3.1)-(3.4) hold, then for
any $k\geq 1$, there are constants $C_{k}\leq\hat{C}_{0}$ and $b_{k}(t)\in
L_{\infty}(-r^{2k}_{0},0;\mathbb{R}^{n^{2}})$ satisfying

$\sup_{t\in(-r^{2k}_{0},0]}|b_{k}(t)|\leq C_{k},$ (3.44)
$\fint_{Q_{r^{k}_{0}}}|\nabla\times u-\overline{(\nabla\times
u)}_{Q_{r^{k}_{0}}}|^{2}dxdt\leq r^{2k\alpha}_{0}$ (3.45)

and

$\displaystyle\sup_{t\in(-r^{2k}_{0},0]}\fint_{B_{r^{k}_{0}}}|u-\overline{u}_{B_{r^{k}_{0}}}(t)-b_{k}(t)x|^{2}dx\leq
r^{2k(1+\alpha)}_{0}.$ (3.46)

Proof. We prove the lemma by induction on $k$. Clearly, it holds as $k=1$ with
$C_{1}=C_{0}$ and $b_{1}(t)=b_{0}(t)$ by Lemma 3.2. Now we suppose there exist
$C_{k}$ and $b_{k}(t)$ such that (3.44)-(3.46) hold for some $k\geq 1$ and
dispose of the case $k+1$ by 7 steps. The constants $\varepsilon_{0}$ and
$\bar{C}_{0}$ will be determined in Step 1 and Step 5 respectively.

_Step 0. Normalization._ For $(x,t)\in Q_{1}$, define

$\tilde{u}(x,t)=r_{0}^{-k(1+\alpha)}u(r^{k}_{0}x,r^{2k}_{0}t),~{}~{}\tilde{p}(x,t)=r_{0}^{-k\alpha}p(r^{k}_{0}x,r^{2k}_{0}t),$
$\tilde{A}(x,t)=A(r^{k}_{0}x,r^{2k}_{0}t),$
$\tilde{f}(x,t)=r_{0}^{-k\alpha}\big{(}f(r^{k}_{0}x,r^{2k}_{0}t)-\overline{f}_{B_{r^{k}_{0}}}(r^{2k}_{0}t)\big{)}~{}~{}\mbox{and}~{}~{}\tilde{g}(x,t)=r_{0}^{-k\alpha}g(r^{k}_{0}x,r^{2k}_{0}t).$

It follows that

$\displaystyle\left\\{\begin{array}[]{ll}\vspace{2mm}\tilde{u}_{t}-\mbox{div}\big{(}\tilde{A}(0,t)\nabla\tilde{u}\big{)}+\nabla\tilde{p}=\mbox{div}\Big{(}\tilde{f}+\big{(}\tilde{A}(x,t)-\tilde{A}(0,t)\big{)}\nabla\tilde{u}\Big{)}~{}~{}\mbox{in}~{}~{}Q_{1},\\\
\vspace{2mm}\mbox{div}~{}\tilde{u}=\tilde{g}~{}~{}\mbox{in}~{}~{}Q_{1}.\end{array}\right.$
(3.49)

By (3.1), (3.3) and (3.4),

$\displaystyle|\tilde{A}(x,t)-\tilde{A}(0,t)|=|A(r^{k}_{0}x,r^{2k}_{0}t)-A(0,r^{2k}_{0}t)|\leq\varepsilon_{0}r_{0}^{k\alpha},$
(3.50) $\displaystyle\begin{aligned}
\fint_{Q_{1}}|\tilde{f}|^{2}dxdt=\fint_{Q_{r^{k}_{0}}}|r_{0}^{-k\alpha}\big{(}f(x,t)-\overline{f}_{B_{r^{k}_{0}}}(t)\big{)}|^{2}dxdt\leq\varepsilon^{2}_{0}\leq\hat{\varepsilon}^{2}_{0}\end{aligned}$
(3.51)

and

$\sup_{t\in(-1,0]}\fint_{B_{1}}|\nabla\tilde{g}|^{q}dx=r^{kq(1-\alpha)}_{0}\sup_{t\in(-r^{2k}_{0},0]}\fint_{B_{r^{k}_{0}}}|\nabla
g(x,t)|^{q}dx\leq 1.$ (3.52)

In view of the induction hypotheses (3.45) and (3.46), we have

$\fint_{Q_{1}}|\nabla\times\tilde{u}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}|^{2}dxdt\leq
1$ (3.53)

and

$\sup_{t\in(-1,0]}\fint_{B_{1}}|\tilde{u}-\alpha(t)-\beta(t)x|^{2}dx\leq 1,$
(3.54)

where

$\alpha(t)=r^{-k(1+\alpha)}_{0}\overline{u}_{B_{r^{k}_{0}}}(r^{2k}_{0}t)~{}~{}\mbox{and}~{}~{}\beta(t)=r^{-k\alpha}_{0}b_{k}(r^{2k}_{0}t).$

_Step 1. Estimates of $\nabla\tilde{u}$._ Applying Lemma 2.3 (i) to
$\tilde{u}-\alpha(t)-\beta(t)x$ with $p=2$,
$\zeta=\overline{(\nabla\times\tilde{u})}_{Q_{1}}-\nabla\times\big{(}\beta(t)x\big{)}$
and $\eta=\overline{\tilde{g}}_{B_{1}}(t)-div\big{(}\beta(t)x\big{)}$, we
obtain, by $\mbox{div}~{}\tilde{u}=\tilde{g}$,

$\displaystyle\begin{aligned}
\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\int_{B_{\frac{3}{4}}}|\nabla\tilde{u}-\beta(t)|^{2}dx=\int_{B_{\frac{3}{4}}}|\nabla\big{(}\tilde{u}-\alpha(t)-\beta(t)x\big{)}|^{2}dx$\mbox{}\hfil\lx@intercol\hskip
8.53581pt\\\ \leq&\
C\Big{(}\int_{B_{1}}|\nabla\times\tilde{u}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}|^{2}dx+\int_{B_{1}}|\tilde{g}-\overline{\tilde{g}}_{B_{1}}(t)|^{2}dx+\int_{B_{1}}|\tilde{u}-\alpha(t)-\beta(t)x|^{2}dx\Big{)}\end{aligned}$

for $t\in(-\frac{9}{16},0]$ and then integrating it with respect to $t$ and
using Poincaré’s inequality,

$\displaystyle\begin{aligned}
&\int_{Q_{\frac{3}{4}}}|\nabla\tilde{u}-\beta(t)|^{2}dxdt\\\ &\leq
C\Big{(}\int_{Q_{1}}|\nabla\times\tilde{u}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}|^{2}dxdt+\int_{Q_{1}}|\nabla\tilde{g}|^{2}dxdt+\int_{Q_{1}}|\tilde{u}-\alpha(t)-\beta(t)x|^{2}dxdt\Big{)},\end{aligned}$

where $C$ depends only on $n$. From (3.52), (3.53) and (3.54), it follows that

$\displaystyle\begin{aligned}
\int_{Q_{\frac{3}{4}}}|\nabla\tilde{u}-\beta(t)|^{2}dxdt\leq C,\end{aligned}$

where $C$ depends only on $n$. It and (3.44) lead to

$\displaystyle\begin{aligned}
\int_{Q_{\frac{3}{4}}}|\nabla\tilde{u}|^{2}dxdt\leq&\
2\int_{Q_{\frac{3}{4}}}|\beta(t)|^{2}dxdt+2\int_{Q_{\frac{3}{4}}}|\nabla\tilde{u}-\beta(t)|^{2}dxdt\\\
&\mbox{}\hskip-51.21504pt\leq
C\sup_{t\in(-\frac{9}{16},0]}|r^{-k\alpha}_{0}b_{k}(r^{2k}_{0}t)|^{2}+C\leq
Cr^{-2k\alpha}_{0}\sup_{t\in(-r^{2k}_{0},0]}|b_{k}(s)|^{2}+C\\\ \leq&\
CC_{k}r^{-2k\alpha}_{0}+C\leq
C\big{(}\hat{C}_{0}r^{-2k\alpha}_{0}+1\big{)}.\end{aligned}$ (3.55)

Set
$\tilde{h}(x,t)=\big{(}\tilde{A}(x,t)-\tilde{A}(0,t)\big{)}\nabla\tilde{u}$
and then (3.49) reads as

$\displaystyle\left\\{\begin{array}[]{ll}\vspace{2mm}\tilde{u}_{t}-\mbox{div}\big{(}\tilde{A}(0,t)\nabla\tilde{u}\big{)}+\nabla\tilde{p}=\mbox{div}\big{(}\tilde{f}+\tilde{h}\big{)}~{}~{}\mbox{in}~{}~{}Q_{1},\\\
\vspace{2mm}\mbox{div}~{}\tilde{u}=\tilde{g}~{}~{}\mbox{in}~{}~{}Q_{1}.\end{array}\right.$
(3.58)

Combining (3.50) and (3.55), we get

$\displaystyle\begin{aligned} \int_{Q_{\frac{3}{4}}}|\tilde{h}|^{2}dxdt=&\
\int_{Q_{\frac{3}{4}}}|\big{(}\tilde{A}(x,t)-\tilde{A}(0,t)\big{)}\nabla\tilde{u}|^{2}dxdt\\\
\leq&C\varepsilon_{0}^{2}r^{2k\alpha}_{0}\big{(}\hat{C}_{0}r^{-2k\alpha}_{0}+1\big{)}\leq
C\varepsilon_{0}^{2}\big{(}\hat{C}_{0}+1\big{)}\leq
C^{*}\hat{\varepsilon}_{0}^{2}\end{aligned}$ (3.59)

as $0<\varepsilon_{0}\leq\hat{\varepsilon}_{0}$ is sufficiently small, where
the constant $C^{*}$ is the same as in (3.17).

_Step 2. Proof of ( 3.45)._ Taking $\nabla\times$ on both sides of the first
equation in (3.58) and by the sane arguments as in Step 2 of the proof of
Lemma 3.2 with (3.51), (3.53) and (3.59) replacing (3.10), (3.9) and (3.17)
respectively, we obtain that

$\displaystyle\fint_{Q_{r_{0}}}|\nabla\times\tilde{u}-\overline{(\nabla\times\tilde{u})}_{Q_{r_{0}}}|^{2}dxdt\leq
2C_{1}\frac{\hat{\varepsilon}_{0}^{2}}{r^{n+2}_{0}}+2C_{2}r^{2}_{0}\leq
r_{0}^{2\alpha},$

where $C_{1}$ and $C_{2}$ are the same as in (3.23) and the last inequality is
derived from (3.41) and (3.42). After rescaling back,

$\displaystyle\begin{aligned} \fint_{Q_{r^{k+1}_{0}}}|\nabla\times
u-\overline{(\nabla\times u)}_{Q_{r^{k+1}_{0}}}|^{2}dxdt\leq
r_{0}^{2(k+1)\alpha}.\end{aligned}$

That is, (3.45) is proved for $k+1$.

_Step 3. Decomposition._ Similar to Step 3 in the proof of Lemma 3.2, we
decompose solution of (3.58) such that
$(\tilde{u},\tilde{p})=(\hat{w},\hat{p}_{\hat{w}})+(\hat{v},\hat{p}_{\hat{v}})$
with

$\displaystyle\left\\{\begin{array}[]{ll}\vspace{2mm}\hat{w}_{t}-\mbox{div}\big{(}\tilde{A}(0,t)\nabla\hat{w}\big{)}+\nabla
p_{\hat{w}}=\mbox{div}\big{(}\tilde{h}+\tilde{f}\big{)}~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\mbox{div}~{}\hat{w}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\hat{w}=0~{}~{}\mbox{on}~{}~{}\partial_{p}Q_{\frac{1}{2}}\end{array}\right.$
(3.63)

and

$\displaystyle\left\\{\begin{array}[]{ll}\vspace{2mm}\hat{v}_{t}-\mbox{div}\big{(}\tilde{A}(0,t)\nabla\hat{v}\big{)}+\nabla
p_{\hat{v}}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\mbox{div}~{}\hat{v}=\tilde{g}~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{2}},\\\
\vspace{2mm}\hat{v}=\tilde{u}~{}~{}\mbox{on}~{}~{}\partial_{p}Q_{\frac{1}{2}}.\end{array}\right.$
(3.67)

_Step 4. Estimates of $\hat{w}$._ Multiplying the first equation in (3.63) by
$\hat{w}$, by the same arguments as in Step 4 of Lemma 3.2 with (3.51) and
(3.59) replacing (3.10) and (3.17) respectively, we have

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{4},0]}\int_{B_{\frac{1}{2}}}|\hat{w}|^{2}dx+\int_{Q_{\frac{1}{2}}}|\nabla\hat{w}|^{2}dxdt\leq
C\int_{Q_{\frac{1}{2}}}|\tilde{h}+\tilde{f}|^{2}dxdt\leq
C\hat{\varepsilon}_{0}^{2}\end{aligned}$ (3.68)

and

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{4},0]}\fint_{B_{r_{0}}}|\hat{w}-\overline{\hat{w}}_{B_{r_{0}}}(t)|^{2}dx\leq\frac{C}{r^{n}_{0}}\sup_{t\in(-\frac{1}{4},0]}\int_{B_{\frac{1}{2}}}|\hat{w}|^{2}dx\leq
C_{3}\frac{\hat{\varepsilon}_{0}^{2}}{r^{n}_{0}},\end{aligned}$ (3.69)

where $C$ depends only on $n$ and $\lambda$, and $C_{3}$ is given by (3.33).

_Step 5. Estimates of $\hat{v}$._ By (3.54), (3.68) and
$\hat{v}=\tilde{u}-\hat{w}$, it follows that

$\displaystyle\begin{aligned}
\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\sup_{t\in(-\frac{1}{4},0]}\|\hat{v}-\alpha(t)-\beta(t)x\|_{L_{2}(B_{\frac{1}{2}})}$\mbox{}\hfil\lx@intercol\hskip
39.83385pt\\\ \leq&\
\sup_{t\in(-\frac{1}{4},0]}\|\tilde{u}-\alpha(t)-\beta(t)x\|_{L_{2}(B_{\frac{1}{2}})}+\sup_{t\in(-\frac{1}{4},0]}\|\hat{w}\|_{L_{2}(B_{\frac{1}{2}})}\leq
C,\end{aligned}$ (3.70)

where $C$ depends only on $n$ and $\lambda$.

Take $\nabla\times$ on both sides of the first equation in (3.67). From
$\nabla\times\nabla p_{\hat{v}}=0$, we deduce

$\displaystyle\begin{array}[]{ll}\vspace{2mm}\big{(}\nabla\times\hat{v}\big{)}_{t}-\mbox{div}\Big{(}\tilde{A}(0,t)\nabla\big{(}\nabla\times\hat{v}\big{)}\Big{)}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{3}}\end{array}$

or, observing that $\overline{(\nabla\times\tilde{u})}_{Q_{1}}$ is a constant,

$\displaystyle\begin{array}[]{ll}\vspace{2mm}\big{(}\nabla\times\hat{v}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\big{)}_{t}-\mbox{div}\Big{(}\tilde{A}(0,t)\nabla\big{(}\nabla\times\hat{v}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\big{)}\Big{)}=0~{}~{}\mbox{in}~{}~{}Q_{\frac{1}{3}}.\end{array}$

By Lemma 2.4 (i), (3.53), (3.68) and
$\nabla\times\hat{v}=\nabla\times\tilde{u}-\nabla\times\hat{w}$,

$\displaystyle\begin{aligned}
&\mbox{}\hskip-22.76228pt\sup_{t\in(-\frac{1}{16},0]}\|\nabla\times\hat{v}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\|_{L_{2}(B_{\frac{1}{4}})}+\sup_{t\in(-\frac{1}{16},0]}\|D(\nabla\times\hat{v})\|_{L_{q}(B_{\frac{1}{4}})}\\\
&\leq
C\left(\|\nabla\times\hat{v}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\|_{L_{\infty}(Q_{\frac{1}{4}})}+\|D(\nabla\times\hat{v})\|_{L_{\infty}(Q_{\frac{1}{4}})}\right)\\\
&\leq
C\|\nabla\times\hat{v}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\|_{L_{2}(Q_{\frac{1}{2}})}=C\|\nabla\times\tilde{u}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}-\nabla\times\hat{w}\|_{L_{2}(Q_{\frac{1}{2}})}\\\
&\leq
C\Big{(}\|\nabla\times\tilde{u}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\|_{L_{2}(Q_{\frac{1}{2}})}+\|\nabla\hat{w}\|_{L_{2}(Q_{\frac{1}{2}})}\Big{)}\leq
C,\end{aligned}$ (3.73)

where $C$ depends only on $n$ and $\lambda$.

Apply Lemma 2.3 (i) to $\hat{v}-\alpha(t)-\beta(t)x$ with $p=2$,
$\zeta=\overline{(\nabla\times\tilde{u})}_{Q_{1}}-\nabla\times\big{(}\beta(t)x\big{)}$
and $\eta=\overline{\tilde{g}}_{B_{1}}(t)-div\big{(}\beta(t)x\big{)}$. Then we
have, by $\mbox{div}~{}\hat{v}=\tilde{g}$ and Poincaré’s inequality,

$\displaystyle\begin{aligned}
&\sup_{t\in(-\frac{1}{36},0]}\|\nabla\hat{v}-\beta(t)\|_{L_{2}(B_{\frac{1}{6}})}\\\
&\leq
C\sup_{t\in(-\frac{1}{16},0]}\Big{(}\|\nabla\times\hat{v}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\|_{L_{2}(B_{\frac{1}{4}})}+\|\tilde{g}-\overline{\tilde{g}}_{B_{1}}(t)\|_{L_{2}(B_{\frac{1}{4}})}+\|\hat{v}-\alpha(t)-\beta(t)x\|_{L_{2}(B_{\frac{1}{4}})}\Big{)}\\\
&\leq
C\sup_{t\in(-\frac{1}{16},0]}\Big{(}\|\nabla\times\hat{v}-\overline{(\nabla\times\tilde{u})}_{Q_{1}}\|_{L_{2}(B_{\frac{1}{4}})}+\|\nabla\tilde{g}(t)\|_{L_{2}(B_{\frac{1}{4}})}+\|\hat{v}-\alpha(t)-\beta(t)x\|_{L_{2}(B_{\frac{1}{4}})}\Big{)}.\end{aligned}$

In view of (3.52), (3.70) and (3.73),

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{36},0]}\|\nabla\hat{v}-\beta(t)\|_{L_{2}(B_{\frac{1}{6}})}\leq
C,\end{aligned}$

which combined with (3.44) leads to

$\displaystyle\begin{aligned}
\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\sup_{t\in(-r_{0}^{2},0]}\Big{(}\fint_{B_{r_{0}}}|\nabla\hat{v}|^{2}dx\Big{)}^{\frac{1}{2}}$\mbox{}\hfil\lx@intercol\hskip
28.45274pt\\\ \leq&\
\sup_{t\in(-r_{0}^{2},0]}\Big{(}\fint_{B_{r_{0}}}|\beta(t)|^{2}dx\Big{)}^{\frac{1}{2}}+\sup_{t\in(-r_{0}^{2},0]}\Big{(}\fint_{B_{r_{0}}}|\nabla\hat{v}-\beta(t)|^{2}dx\Big{)}^{\frac{1}{2}}\\\
\leq&\
\sup_{t\in(-r_{0}^{2},0]}|r^{-k\alpha}_{0}b_{k}(r^{2k}_{0}t)|+\frac{C}{r^{\frac{n}{2}}_{0}}\sup_{t\in(-r_{0}^{2},0]}\Big{(}\int_{B_{\frac{1}{6}}}|\nabla\hat{v}-\beta(t)|^{2}dx\Big{)}^{\frac{1}{2}}\\\
\leq&\
r^{-k\alpha}_{0}\sup_{t\in(-r_{0}^{2+2k},0]}|b_{k}(s)|+\bar{C}_{0}r^{-\frac{n}{2}}_{0}\leq
C_{k}r^{-k\alpha}_{0}+\bar{C}_{0}r^{-\frac{n}{2}}_{0},\end{aligned}$ (3.74)

where $C$ and $\bar{C}_{0}$ depend only on $n$ and $\lambda$.

Apply Lemma 2.3 (ii) to $\hat{v}-\alpha(t)-\beta(t)x$ and then by (3.52),
(3.70) and (3.73),

$\displaystyle\begin{aligned}
\sup_{t\in(-\frac{1}{36},0]}\|D^{2}\hat{v}\|_{L_{q}(B_{\frac{1}{6}})}\leq&\
C\sup_{t\in(-\frac{1}{16},0]}\Big{(}\|D\nabla\times\hat{v}\|_{L_{q}(B_{\frac{1}{4}})}+\|\nabla\tilde{g}\|_{L_{q}(B_{\frac{1}{4}})}\\\
&\mbox{}\hskip
85.35826pt+\|\hat{v}-\alpha(t)-\beta(t)x\|_{L_{2}(B_{\frac{1}{4}})}\Big{)}\leq
C.\end{aligned}$

It follows that

$\displaystyle\begin{aligned}
\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|\hat{v}-\overline{\hat{v}}_{B_{r_{0}}}(t)-\overline{\nabla\hat{v}}_{B_{r_{0}}}(t)x|^{2}dx$\mbox{}\hfil\lx@intercol\hskip
45.5244pt\\\ \leq&\
Cr^{4}_{0}\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|D^{2}\hat{v}|^{2}dx\leq
Cr^{4}_{0}\sup_{t\in(-r_{0}^{2},0]}\Big{(}\fint_{B_{r_{0}}}|D^{2}\hat{v}|^{q}dx\Big{)}^{\frac{2}{q}}\\\
\leq&\
Cr^{4-\frac{2n}{q}}_{0}\sup_{t\in(-r_{0}^{2},0]}\|D^{2}\hat{v}\|^{2}_{L_{q}(B_{\frac{1}{6}})}\leq
C_{4}r^{4-\frac{2n}{q}}_{0},\end{aligned}$ (3.75)

where $C_{4}$ is the same constant as in (3.38).

_Step 6. Proof of ( 3.44) and (3.46)._ By (3.69) and (3.75), we obtain

$\displaystyle\begin{aligned}
\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|\tilde{u}-\overline{\tilde{u}}_{B_{r_{0}}}(t)-\overline{\nabla\hat{v}}_{B_{r_{0}}}(t)x|^{2}dx$\mbox{}\hfil\lx@intercol\hskip
34.1433pt\\\ \leq&\
2\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|\hat{v}-\overline{\hat{v}}_{B_{r_{0}}}(t)-\overline{\nabla\hat{v}}_{B_{r_{0}}}(t)x|^{2}dx+2\sup_{t\in(-r_{0}^{2},0]}\fint_{B_{r_{0}}}|\hat{w}-\overline{\hat{w}}_{B_{r_{0}}}(t)|^{2}dx\\\
\leq&\
2C_{4}r^{4-\frac{2n}{q}}_{0}+2C_{3}\frac{\hat{\varepsilon}_{0}^{2}}{r^{n}_{0}}\leq
r_{0}^{2+2\alpha},\end{aligned}$

where the last inequality is derived from (3.41) and (3.42). Rewriting the
above inequality,

$\displaystyle\sup_{t\in(-r_{0}^{2(k+1)},0]}\fint_{B_{r^{k+1}_{0}}}|u-\overline{u}_{B_{r^{k+1}_{0}}}(t)-b_{k+1}(t)x|^{2}dx\leq
r^{2(k+1)(1+\alpha)}_{0}$

with

$\displaystyle
b_{k+1}(t)=r^{k\alpha}_{0}\overline{\nabla\hat{v}}_{B_{r_{0}}}\Big{(}\frac{t}{r^{2k}_{0}}\Big{)}.$

By (3.74), we have

$\displaystyle\begin{aligned} \sup_{t\in(-r_{0}^{2(k+1)},0]}|b_{k+1}(t)|\leq&\
\sup_{t\in(-r_{0}^{2(k+1)},0]}r_{0}^{k\alpha}\Big{(}\fint_{B_{r_{0}}}|\nabla\hat{v}(x,\frac{t}{r^{2k}_{0}})|^{2}dx\Big{)}^{\frac{1}{2}}\\\
=&\
\sup_{t\in(-r_{0}^{2},0]}r_{0}^{k\alpha}\Big{(}\fint_{B_{r_{0}}}|\nabla\hat{v}|^{2}dx\Big{)}^{\frac{1}{2}}\leq
r_{0}^{k\alpha}\Big{(}C_{k}r^{-k\alpha}_{0}+\frac{\bar{C}_{0}}{r^{\frac{n}{2}}_{0}}\Big{)}\\\
\leq&\
C_{k}+\bar{C}_{0}r^{-\frac{n}{2}}_{0}r_{0}^{k\alpha}:=C_{k+1}.\end{aligned}$

Then from induction, we deduce

$C_{k+1}=C_{0}+{\bar{C}}_{0}r_{0}^{-\frac{n}{2}}\sum_{j=1}^{k}r_{0}^{j\alpha}.$

In view of (3.43), we have $C_{k+1}\leq\hat{C}_{0}$. That is, (3.44) and
(3.46) are obtained for $k+1$ and the proof is completed. ∎

Proof of Theorem 3.1. For any $0<r<1$, let $k\geq 1$ such that

$r^{k+1}_{0}\leq r\leq r^{k}_{0}.$ (3.76)

By (3.45), we have

$\displaystyle\fint_{Q_{r}}|\nabla\times u-\overline{(\nabla\times
u)}_{Q_{r}}|^{2}dxdt$ $\displaystyle\mbox{}\hskip 28.45274pt\leq
2\frac{|Q_{r^{k}_{0}}|}{|Q_{r}|}\fint_{Q_{r^{k}_{0}}}|\nabla\times
u-\overline{(\nabla\times u)}_{Q_{r^{k}_{0}}}|^{2}dxdt\leq
Cr^{2k\alpha}_{0}\leq Cr^{2\alpha}$

and hence (3.6) holds.

For any $t\in(-r^{2}_{0},0)$, define

$a(t)=\overline{u}_{B_{r^{k}_{0}}}(t)~{}\mbox{as}~{}t\in(-r^{2k}_{0},-r^{2(k+1)}_{0}]$

and

$b(t)=b_{k}(t)~{}\mbox{as}~{}t\in(-r^{2k}_{0},-r^{2(k+1)}_{0}]$

for $k\geq 1$, where $b_{k}$ is given by Lemma 3.3.

From (3.46), we deduce

$\displaystyle\mbox{}\hskip-45.52458pt\sup_{t\in(-r^{2(k+1)}_{0},0]}\fint_{B_{r_{0}^{k+1}}}|\overline{u}_{B_{r^{k+1}_{0}}}(t)-\overline{u}_{B_{r^{k}_{0}}}(t)+b_{k+1}(t)x-b_{k}(t)x|^{2}dx$
$\displaystyle\leq
C\Bigg{\\{}\sup_{t\in(-r^{2k}_{0},0]}\fint_{B_{r_{0}^{k}}}|u-\overline{u}_{B_{r_{0}^{k}}}(t)-b_{k}(t)x|^{2}dx$
$\displaystyle\mbox{}\hskip
56.9055pt+\sup_{t\in(-r^{2(k+1)}_{0},0]}\fint_{B_{r_{0}^{k+1}}}|u-\overline{u}_{B_{r_{0}^{k+1}}}(t)-b_{k+1}(t)x|^{2}dx\Bigg{\\}}$
$\displaystyle\leq Cr^{2k(1+\alpha)}_{0}$

and then by Lemma 2.2, we have

$\displaystyle\sup_{t\in(-r^{2(k+1)}_{0},0]}|\overline{u}_{B_{r^{k}_{0}}}(t)-\overline{u}_{B_{r^{k+1}_{0}}}(t)|\leq$
$\displaystyle\ Cr^{k(1+\alpha)}_{0}$

and

$\sup_{t\in(-r^{2(k+1)}_{0},0]}|b_{k}(t)-b_{k+1}(t)|\leq Cr^{k\alpha}_{0}.$

Therefore we can and we do define

$a(0)=\lim_{k\rightarrow\infty}\overline{u}_{B_{r^{k}_{0}}}(0)~{}~{}\mbox{and}~{}~{}b(0)=\lim_{k\rightarrow\infty}b_{k}(0).$

Since

$\displaystyle\sup_{t\in(-r^{2k}_{0},0]}|a(t)-\overline{u}_{B_{r^{k}_{0}}}(t)|=\sup_{t\in(-r^{2(k+1)}_{0},0]}|a(t)-\overline{u}_{B_{r^{k}_{0}}}(t)|$
$\displaystyle\leq$ $\displaystyle\
\sup_{t\in(-r^{2(k+1)}_{0},0]}\Big{(}|a(t)-\overline{u}_{B_{r^{k+1}_{0}}}(t)|+|\overline{u}_{B_{r^{k}_{0}}}(t)-\overline{u}_{B_{r^{k+1}_{0}}}(t)|\Big{)}$
$\displaystyle\leq$ $\displaystyle\
\sup_{t\in(-r^{2(k+1)}_{0},0]}|a(t)-\overline{u}_{B_{r^{k+1}_{0}}}(t)|+Cr^{k(1+\alpha)}_{0},$

we have, by repeating the above inequalities,

$\sup_{t\in(-r^{2k}_{0},0]}|a(t)-\overline{u}_{B_{r^{k}_{0}}}(t)|\leq
C\sum_{j=k}^{\infty}r_{0}^{j(1+\alpha)}\leq Cr_{0}^{k(1+\alpha)}$

and similarly,

$\sup_{t\in(-r^{2k}_{0},0]}|b(t)-b_{k}(t)|\leq
C\sum_{j=k}^{\infty}r_{0}^{j\alpha}\leq Cr_{0}^{k\alpha},$

where $C$ depends only on $n$, $\lambda$, $\alpha$ and $q$. By (3.46) again,

$\displaystyle\sup_{t\in(-r^{2k}_{0},0]}\fint_{B_{r_{0}^{k}}}|u-a(t)-b(t)x|^{2}dx$
$\displaystyle\leq$ $\displaystyle\
3\sup_{t\in(-r^{2k}_{0},0]}\Big{(}\fint_{B_{r^{k}_{0}}}|u-\overline{u}_{B_{r^{k}_{0}}}(t)-b_{k}(t)x|^{2}dx+|a(t)-\overline{u}_{B_{r^{k}_{0}}}(t)|^{2}+|b(t)-b_{k}(t)|^{2}r_{0}^{2k}\Big{)}$
$\displaystyle\leq$ $\displaystyle\ Cr_{0}^{2k(1+\alpha)}.$

For any $0<r\leq r_{0}$, let $k$ be given by (3.76) and it follows that

$\displaystyle\sup_{t\in(-r^{2},0]}\fint_{B_{r}}|u-a(t)-b(t)x|^{2}dx\leq$
$\displaystyle\
r_{0}^{-n}\sup_{t\in(-r^{2k}_{0},0]}\fint_{B_{r_{0}^{k}}}|u-a(t)-b(t)x|^{2}dx$
$\displaystyle\leq$ $\displaystyle\ Cr_{0}^{2k(1+\alpha)}\leq
Cr^{2(1+\alpha)}.$

Thus (3.7) is proved.

Finally, we show the boundedness of $a(t)$ and $b(t)$. Actually, by (3.44), we
obtain

$\sup_{t\in(-r^{2}_{0},0]}|b(t)|\leq\sup_{k}\sup_{t\in(-r^{2k}_{0},-r^{2(k+1)}_{0}]}|b_{k}(t)|\leq\sup_{k}C_{k}\leq\hat{C}_{0}$

and from (3.2), we see

$\displaystyle\sup_{t\in(-r^{2}_{0},0]}|a(t)|=$ $\displaystyle\
\sup_{t\in(-r^{2}_{0},0]}|\overline{u}_{B_{r_{0}}}(t)|+\sup_{t\in(-r^{2}_{0},0]}|a(t)-\overline{u}_{B_{r_{0}}}(t)|$
$\displaystyle\leq$ $\displaystyle\
Cr^{-\frac{n}{2}}_{0}\sup_{t\in(-r^{2}_{0},0]}\Big{(}\int_{B_{1}}|u|^{2}dx\Big{)}^{\frac{1}{2}}+Cr_{0}^{1+\alpha}.$

The proof of Theorem 3.1 is complete. ∎

Proof of Corollary 1.2. Let
$R=\min\\{\mbox{dist}(\Omega^{\prime},\partial\Omega),\sqrt{T-T^{\prime}}\\}$
and $q=\frac{2n}{1-\alpha}$. For any
$(x_{0},t_{0})\in\Omega^{\prime}_{T^{\prime}}$, we have (1.2), (1.3) and (1.4)
hold with $M_{1}=\|f\|^{2}_{C_{x}^{\alpha}(\Omega_{T})}$ and $M_{2}=\|\nabla
g\|_{L_{\infty}(\Omega_{T})}^{q}$. Then by Theorem 1.1, there exists $\sigma$
depending only on $n$, $\lambda$, $\alpha$ and $M_{0}$ such that (1.5) holds
for any $0<r\leq r_{0}$ with $r_{0}=\sigma R$. From Lemma 2.1 (i), we obtain
the estimate of $[\nabla\times
u]_{C^{\alpha,\frac{\alpha}{2}}(\overline{\Omega^{\prime}_{T^{\prime}}})}$.
Then the desired estimate for $\|\nabla\times
u\|_{C^{\alpha,\frac{\alpha}{2}}(\overline{\Omega^{\prime}_{T^{\prime}}})}$ in
(1.7) follows from the boundedness of $\|\nabla\times u\|_{L_{2}(\Omega_{T})}$
immediately.

Now we turn to the estimate for $\|\nabla
u\|_{C_{x}^{\alpha}(\overline{\Omega^{\prime}_{T^{\prime}}})}$. By Theorem
1.1, there exist $a(x_{0},t)\in
L_{\infty}(t_{0}-r_{0}^{2},t_{0};\mathbb{R}^{n})$ and $b(x_{0},t)\in
L_{\infty}(t_{0}-r_{0}^{2},t_{0};\mathbb{R}^{n^{2}})$ such that (1.6) holds
for any $0<r\leq r_{0}$ with $a(x_{0},t)$ and $b(x_{0},t)$ replacing $a(t)$
and $b(t)$ respectively. Set $t=t_{0}$ in the obtained (1.6) and then

$\displaystyle\left(\fint_{B_{r}(x_{0})}|u(x,t_{0})-a(x_{0},t_{0})-b(x_{0},t_{0})x|^{2}dx\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\
Cr^{1+\alpha}\Big{(}\|u\|_{L_{\infty}(-T,0;L_{2}(\Omega))}+\|\nabla\times
u\|_{L_{2}(\Omega_{T})}+\|f\|_{C_{x}^{\alpha}(\Omega_{T})}+\|\nabla
g\|_{L_{\infty}(\Omega_{T})}\Big{)}$

for any $0<r\leq r_{0}$, where $C$ depends only on $n$, $\lambda$, $\alpha$
and $M_{0}$. Then by Lemma 2.1 (ii) and the boundedness of
$\|u\|_{L_{\infty}(-T,0;L_{2}(\Omega))}$, we obtain the desired estimate of
$\|\nabla u\|_{C_{x}^{\alpha}(\overline{\Omega^{\prime}_{T^{\prime}}})}$
clearly. This gives (1.7) completely. ∎

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