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<top_article>
<mrnumber>MR3157151</mrnumber>
<author>Yuko ENOMOTO and Yoshihiro SHIBATA</author>
<author_utf8>Yuko ENOMOTO and Yoshihiro SHIBATA</author_utf8>
<title>On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow</title>
<journal>Funkcialaj Ekvacioj. Serio Internacia</journal>
<volume>56</volume>
<year>2013</year>
<page>441--505</page>
<url_pdf>http://fe.math.kobe-u.ac.jp/FE/FullPapers/56-3/56_441.pdf</url_pdf>
<mathsci_link>http://www.ams.org/mathscinet-getitem?mr=MR3157151</mathsci_link>
<abstract>In this paper, we prove the $\mathcal{R}$-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].</abstract>
<keywords>Compressible viscous fluid, Navier-Stokes equations, Stokes equations, $\mathcal{R}$-sectoriality, General domain, Analytic semigroup, Maximal $L_p$-$L_q$ regularity, Exponential stability, Local in time unique existence theorem, Global in time unique existence theorem.</keywords>
<subject>35Q30, 76D07.</subject>
<fesi_info>
  <FILE>56-441</FILE>
  <YEAR>2013</YEAR>
  <TITLE>On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow</TITLE>
  <AUTHOR>Yuko ENOMOTO and Yoshihiro SHIBATA</AUTHOR>
  <AUTHOR_utf8>Yuko ENOMOTO and Yoshihiro SHIBATA</AUTHOR_utf8>
</fesi_info>

<references>


<article>
<bibitem>1</bibitem>
<author>Abels, H.; Terasawa, Y.</author>
<title>On Stokes operators with variable viscosity in bounded and unbounded domains</title>
<journal>Math. Ann.</journal>
<vol>344</vol>
<year>2009</year>
<page>381-429</page>
<mr>MR2495776</mr>
</article>


<article>
<bibitem>2</bibitem>
<author>Agmon, S.</author>
<title>On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems</title>
<journal>Comm. Pure Appl. Math.</journal>
<vol>15</vol>
<year>1962</year>
<page>119-147</page>
<mr>MR0147774</mr>
</article>


<article>
<bibitem>3</bibitem>
<author>Agmon, S.; Douglis, A.; Nirenberg, L.</author>
<title>Estimates near the boundary for solutions of elliptic partial 
differential equations satisfying general boundary conditions, I</title>
<journal>Comm. Pure Appl. Math.</journal>
<vol>12</vol>
<year>1959</year>
<page>623-727</page>
<mr>MR0125307</mr>
</article>


<article>
<bibitem>4</bibitem>
<author>Agmon, S.; Douglis, A.; Nirenberg, L.</author>
<title>Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II</title>
<journal>Comm. Pure Appl. Math.</journal>
<vol>17</vol>
<year>1964</year>
<page>35-92</page>
<mr>MR0162050</mr>
</article>


<article>
<bibitem>5</bibitem>
<author>Agranovich, M.; Denk, R.; Faiermann, M.</author>
<title>Weakly smooth nonselfadjoint spectral elliptic boundary problems</title>
<journal>Math. Top., 14, Akademie Verlag, Berlin</journal>
<vol></vol>
<year>1997</year>
<page>pp. 138-199</page>
<mr>MR1608279</mr>
</article>


<article>
<bibitem>6</bibitem>
<author>Agranovich, M.; Vishik, M. I.</author>
<title>Elliptic problems with a parameter and parabolic problems of general type</title>
<journal>Russian Math. Surveys</journal>
<vol>19</vol>
<year>1964</year>
<page>53-157</page>
<mr>MR0192188</mr>
</article>


<article>
<bibitem>7</bibitem>
<author>Amann, H.</author>
<title>Dynamic theory of quasilinear parabolic equations-II, Reaction-diffusion systems</title>
<journal>Differential Integral Equations</journal>
<vol>3</vol>
<year>1990</year>
<page>13-75</page>
<mr>MR1014726</mr>
</article>


<book>
<bibitem>8</bibitem>
<author>Amann, H.</author>
<booktitle>Linear and Quasilinear Parabolic Problems, Vol. I</booktitle>
<publisher>Monographs in Mathematics, 89, Birkh&#228;user Boston, Inc., Boston, MA</publisher>
<year>1995</year>
<mr>MR1345385</mr>
</book>


<article>
<bibitem>9</bibitem>
<author>Bourgain, J.</author>
<title>Vector-valued singular integrals and the $H^1$-BMO duality</title>
<journal>Monogr. Textbooks Pure Appl. Math., 98, Dekker, New York</journal>
<vol></vol>
<year>1986</year>
<page>pp. 1-19</page>
<mr>MR0830227</mr>
</article>


<article>
<bibitem>10</bibitem>
<author>Charve, F.; Danchin, R.</author>
<title>A global existence result for the compressible Navier-Stokes equations in the critical $L_p$ framework</title>
<journal>Arch. Ration. Mech. Anal.</journal>
<vol>198</vol>
<year>2010</year>
<page>233-271</page>
<mr>MR2679372</mr>
</article>


<article>
<bibitem>11</bibitem>
<author>Cl&#233;ment, Ph.; Pr&#252;\ss, J.</author>
<title>An operator-valued transference principle and maximal regularity on vector-valued $L_p$-spaces</title>
<journal>Lecture Notes in Pure and Appl. Math., 215, Dekker, New York</journal>
<vol></vol>
<year>2001</year>
<page>pp. 67-87</page>
<mr>MR1816437</mr>
</article>


<book>
<bibitem>12</bibitem>
<author>Denk, R.; Hieber, M.; Pr&#252;&#223;, J.</author>
<booktitle>$\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type</booktitle>
<publisher>Mem. Amer. Math. Soc., 166, no. 788</publisher>
<year>2003</year>
<mr>MR2006641</mr>
</book>


<article>
<bibitem>13</bibitem>
<author>Enomoto, Y.; Shibata, Y.</author>
<title>About compressible viscous fluids in a 2-dimensional exterior domain</title>
<journal>Oper. Theory Adv. Appl., 221, Birkh&#228;user/Springer Basel AG, Basel</journal>
<vol></vol>
<year>2012</year>
<page>pp. 305-321</page>
<mr>MR2954073</mr>
</article>


<article>
<bibitem>14</bibitem>
<author>Farwig, R.; Kozono, H.; Sohr, H.</author>
<title>An $L^q$-approach to Stokes and Navier-Stokes equations in general domains</title>
<journal>Acta Math.</journal>
<vol>195</vol>
<year>2005</year>
<page>21-53</page>
<mr>MR2233684</mr>
</article>


<article>
<bibitem>15</bibitem>
<author>Farwig, R.; Sohr, H.</author>
<title>Generalized resolvent estimates for the Stokes system in bounded and unbounded domains</title>
<journal>J. Math. Soc. Japan</journal>
<vol>46</vol>
<year>1994</year>
<page>607-643</page>
<mr>MR1291109</mr>
</article>


<article>
<bibitem>16</bibitem>
<author>Graffi, D.</author>
<title>Il teorema di unicit&#224; nella dinamica dei fluidi compressibili
</title>
<journal>J. Rational Mech. Anal.</journal>
<vol>2</vol>
<year>1953</year>
<page>99-106</page>
<mr>MR0052270</mr>
</article>


<article>
<bibitem>17</bibitem>
<author>Heywood, J.</author>
<title>The Navier-Stokes equations: On the existence, regularity and decay of solutions</title>
<journal>Indiana Univ. Math. J.</journal>
<vol>29</vol>
<year>1980</year>
<page>639-681</page>
<mr>MR0589434</mr>
</article>


<article>
<bibitem>18</bibitem>
<author>H&#246;rmander, L.</author>
<title>Estimates for translation invariant operators in $L^p$ spaces</title>
<journal>Acta Math.</journal>
<vol>104</vol>
<year>1960</year>
<page>93-140</page>
<mr>MR0121655</mr>
</article>


<article>
<bibitem>19</bibitem>
<author>Itaya, N.</author>
<title>On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluids</title>
<journal>K&#333;dai Math. Sem. Rep.</journal>
<vol>23</vol>
<year>1971</year>
<page>60-120</page>
<mr>MR0283426</mr>
</article>


<article>
<bibitem>20</bibitem>
<author>Itaya, N.</author>
<title>On the initial value problem of the motion of compressible viscous fluid, especially in the problem of uniqueness</title>
<journal>J. Math. Kyoto Univ.</journal>
<vol>16</vol>
<year>1976</year>
<page>413-427</page>
<mr>MR0421305</mr>
</article>


<article>
<bibitem>21</bibitem>
<author>Ishihara, Y.; Kagei, Y.</author>
<title>Large time behavior of the semigroup on $L_p$ spaces associated with the linearized compressible Navier-Stokes equation in a cylindrical domain</title>
<journal>J. Differential Equations</journal>
<vol>248</vol>
<year>2010</year>
<page>252-286</page>
<mr>MR2558166</mr>
</article>


<fearticle>
<bibitem>22</bibitem>
<author>Kagei, Y.</author>
<title>Resolvent estimates for the linearized compressible Navier-Stokes equations in an infinite layer</title>
<journal>Funkcial. Ekvac.</journal>
<vol>50</vol>
<year>2007</year>
<page>287-337</page>
<mr>MR2351216</mr>
<feart>2351216</feart>
</fearticle>


<article>
<bibitem>23</bibitem>
<author>Kagei, Y.</author>
<title>Asymptotic behavior of the semigroup associated with the linearized compressible Navier-Stokes equation in an infinite layer</title>
<journal>Publ. Res. Inst. Math. Sci.</journal>
<vol>43</vol>
<year>2007</year>
<page>763-794</page>
<mr>MR2361795</mr>
</article>


<article>
<bibitem>24</bibitem>
<author>Kagei, Y.</author>
<title>Large time behavior of solutions to the compressible Navier-Stokes equation in an infinite layer</title>
<journal>Hiroshima Math. J.</journal>
<vol>38</vol>
<year>2008</year>
<page>95-124</page>
<mr>MR2397381</mr>
</article>


<article>
<bibitem>25</bibitem>
<author>Kagei, Y.; Nukumizu, T.</author>
<title>Asymptotic behavior of solutions to the compressible Navier-Stokes equation in a cylindrical domain</title>
<journal>Osaka J. Math.</journal>
<vol>45</vol>
<year>2008</year>
<page>987-1026</page>
<mr>MR2493967</mr>
</article>


<article>
<bibitem>26</bibitem>
<author>Kagei, Y.</author>
<title>Decay estimates on solutions of the linearized compressible Navier-Stokes equation around a Poiseuille type flow</title>
<journal>J. Math-for-Ind.</journal>
<vol>2A</vol>
<year>2010</year>
<page>39-56</page>
<mr>MR2639364</mr>
</article>


<article>
<bibitem>27</bibitem>
<author>Kagei, Y.; Kobayashi, T.</author>
<title>On large-time behavior of solutions to the compressible Navier-Stokes equations in the half space in $\mathbb{R}^3$</title>
<journal>Arch. Ration. Mech. Anal.</journal>
<vol>165</vol>
<year>2002</year>
<page>89-159</page>
<mr>MR1946756</mr>
</article>


<article>
<bibitem>28</bibitem>
<author>Kagei, Y.; Kobayashi, T.</author>
<title>Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space</title>
<journal>Arch. Ration. Mech. Anal.</journal>
<vol>177</vol>
<year>2005</year>
<page>231-330</page>
<mr>MR2188049</mr>
</article>


<article>
<bibitem>29</bibitem>
<author>Matsumura, A.; Nishida, T.</author>
<title>The initial value problem for the equations of motion of viscous and heat-conductive gases</title>
<journal>J. Math. Kyoto Univ.</journal>
<vol>20</vol>
<year>1980</year>
<page>67-104</page>
<mr>MR0564670</mr>
</article>


<article>
<bibitem>30</bibitem>
<author>Matsumura, A.; Nishida, T.</author>
<title>Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids</title>
<journal>Comm. Math. Phys.</journal>
<vol>89</vol>
<year>1983</year>
<page>445-464</page>
<mr>MR0713680</mr>
</article>


<article>
<bibitem>31</bibitem>
<author>Mikhlin, S. G.</author>
<title>Fourier integrals and multiple singular integrals (Russian)</title>
<journal>Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr.</journal>
<vol>12</vol>
<year>1957</year>
<page>143-145</page>
<mr>MR0088603</mr>
</article>


<article>
<bibitem>32</bibitem>
<author>Nash, J.</author>
<title>Le probl&#232;me de Cauchy pour les &#233;quations diff&#233;rentielles d'un fluide g&#233;n&#233;ral</title>
<journal>Bull. Soc. Math. France</journal>
<vol>90</vol>
<year>1962</year>
<page>487-497</page>
<mr>MR0149094</mr>
</article>


<article>
<bibitem>33</bibitem>
<author>Serrin, J.</author>
<title>On the uniqueness of compressible fluid motion</title>
<journal>Arch. Rational Mech. Anal.</journal>
<vol>3</vol>
<year>1959</year>
<page>271-288</page>
<mr>MR0106646</mr>
</article>


<article>
<bibitem>34</bibitem>
<author>Shibata, Y.; Shimizu, S.</author>
<title>A decay property of the Fourier transform and its application to the Stokes problem</title>
<journal>J. Math. Fluid Mech.</journal>
<vol>3</vol>
<year>2001</year>
<page>213-230</page>
<mr>MR1860123</mr>
</article>


<article>
<bibitem>35</bibitem>
<author>Shibata, Y.; Shimizu, S.</author>
<title>On the $L_p$-$L_q$ maximal regularity of the Neumann problem for the Stokes equations in a bounded domain</title>
<journal>J. Reine Angew. Math.</journal>
<vol>615</vol>
<year>2008</year>
<page>157-209</page>
<mr>MR2384339</mr>
</article>


<article>
<bibitem>36</bibitem>
<author>Shibata, Y.; Shimizu, S.</author>
<title>On a free boundary problem for the Navier-Stokes equations</title>
<journal>Differential Integral Equations</journal>
<vol>20</vol>
<year>2007</year>
<page>241-276</page>
<mr>MR2293985</mr>
</article>


<article>
<bibitem>37</bibitem>
<author>Shibata, Y.; Tanaka, K.</author>
<title>On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion</title>
<journal>Math. Methods Appl. Sci.</journal>
<vol>27</vol>
<year>2004</year>
<page>1579-1606</page>
<mr>MR2077445</mr>
</article>


<article>
<bibitem>38</bibitem>
<author>Solonnikov, V. A.</author>
<title>On boundary problems for linear parabolic systems of differential equations of general type</title>
<journal>Trudy Mat. Inst. Steklov.</journal>
<vol>83</vol>
<year>1965</year>
<page>3-163 (in Russian); English transl.: Proc. Steklov Inst. Math., 83 (1967), 1-184</page>
<mr>MR0211083</mr>
</article>


<article>
<bibitem>39</bibitem>
<author>Solonnikov, V. A.</author>
<title>On the solvability of the initial boundary value problem for the equations of the motion of a viscous compressible fluid</title>
<journal>Zap. Nau&#269;n. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)</journal>
<vol>56</vol>
<year>1976</year>
<page>128-142 (in Russian); J. Soviet Math., 14 (1980), 1120-1133</page>
<mr>MR0481666</mr>
</article>


<book>
<bibitem>40</bibitem>
<author>Stein, E. M.</author>
<booktitle>Singular Integrals and Differentiability Properties of Functions</booktitle>
<publisher>Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ</publisher>
<year>1970</year>
<mr>MR0290095</mr>
</book>


<article>
<bibitem>41</bibitem>
<author>Stewart, B.</author>
<title>Generation of analytic semigroups by strongly elliptic operators under general boundary conditions</title>
<journal>Trans. Amer. Math. Soc.</journal>
<vol>259</vol>
<year>1980</year>
<page>299-310</page>
<mr>MR0561838</mr>
</article>


<article>
<bibitem>42</bibitem>
<author>Str&#246;hmer, G.</author>
<title>About the resolvent of an operator from fluid dynamics</title>
<journal>Math. Z.</journal>
<vol>194</vol>
<year>1987</year>
<page>183-191</page>
<mr>MR0876229</mr>
</article>


<article>
<bibitem>43</bibitem>
<author>Str&#246;hmer, G.</author>
<title>About a certain class of parabolic-hyperbolic systems of differential equations</title>
<journal>Analysis</journal>
<vol>9</vol>
<year>1989</year>
<page>1-39</page>
<mr>MR0998166</mr>
</article>


<article>
<bibitem>44</bibitem>
<author>Str&#246;hmer, G.</author>
<title>About compressible viscous fluid flow in a bounded region</title>
<journal>Pacific J. Math.</journal>
<vol>143</vol>
<year>1990</year>
<page>359-375</page>
<mr>MR1051082</mr>
</article>


<book>
<bibitem>45</bibitem>
<author>Tanabe, H.</author>
<booktitle>Functional Analytic Methods for Partial Differential Equations</booktitle>
<publisher>Monographs and Textbooks in Pure and Applied Mathematics, 204, Marchel Dekker, Inc., New York</publisher>
<year>1997</year>
<mr>MR1413304</mr>
</book>


<article>
<bibitem>46</bibitem>
<author>Tani, A.</author>
<title>On the first initial-boundary value problem of compressible viscous fluid motion</title>
<journal>Publ. Res. Inst. Math. Sci.</journal>
<vol>13</vol>
<year>1977</year>
<page>193-253</page>
<mr></mr>
</article>


<article>
<bibitem>47</bibitem>
<author>Valli, A.</author>
<title>An existence theorem for compressible viscous fluids</title>
<journal>Ann. Mat. Pura Appl. (4)</journal>
<vol>130</vol>
<year>1982</year>
<page>197-213</page>
<mr>MR0663971</mr>
</article>


<article>
<bibitem>48</bibitem>
<author>Valli, A.; Zaja&#231;zkowski, W.</author>
<title>Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case</title>
<journal>Comm. Math. Phys.</journal>
<vol>103</vol>
<year>1986</year>
<page>259-296</page>
<mr>MR0826865</mr>
</article>


<article>
<bibitem>49</bibitem>
<author>Volevich, L. R.</author>
<title>Solvability of boundary value problems for general elliptic systmes</title>
<journal>Mat. Sb. (N.S.)</journal>
<vol>68 (110)</vol>
<year>1965</year>
<page>373-416 (in Russian); English transl. Amer. Math. Soc. Transl., Ser. 2., 67 (1968), 182-225</page>
<mr>MR0192191</mr>
</article>


<article>
<bibitem>50</bibitem>
<author>Vol'pert, A. I.; Hudjaev, S. I.</author>
<title>On the Cauchy problem for composite systems of nonlinear differential equations</title>
<journal>Math. USSR-Sb.</journal>
<vol>16</vol>
<year>1972</year>
<page>514-544</page>
<mr>MR0390528</mr>
</article>


<article>
<bibitem>51</bibitem>
<author>Weis, L.</author>
<title>Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity</title>
<journal>Math. Ann.</journal>
<vol>319</vol>
<year>2001</year>
<page>735-758</page>
<mr>MR1825406</mr>
</article>


</references>
</top_article>
