Publication Cover
Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 2
320
Views
7
CrossRef citations to date
0
Altmetric
Articles

Local well-posedness in critical spaces for the compressible MHD equations

&
Pages 239-269 | Received 19 Nov 2013, Accepted 29 Mar 2014, Published online: 06 May 2014

References

  • Nash J. Le problème de cauchy pour les équations différentielles d’un fluide général [The Cauchy problem for the differential equations of a general fluid]. Bull. Soc. Math. France. 1962;90:487–497.
  • Itaya N. On initial value problem of the motion of compressible viscous fluid, especially on the problem of uniqueness. J. Math. Kyoto Univ. 1976;16:413–427.
  • Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 1980;20:67–104.
  • Chen Q, Miao C, Zhang Z. On the well-posedness for the viscous shallow water equations. SIAM J. Math. Anal. 2008;40:443–474.
  • Danchin R. Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Par. Differ. Equ. 2001;26:1183–1233.
  • Danchin R. Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Ration. Mech. Anal. 2001;160:1–39.
  • Danchin R. Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 2000;141:579–614.
  • Danchin R. On the uniqueness in critical spaces for compressible Navier–Stokes equations. NoDEA Nonlinear Differ. Equ. Appl. 2005;12:111–128.
  • Danchin R. Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Comm. Par. Differ. Equ. 2007;32:1373–1397.
  • Danchin R. Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A. 2003;133:1311–1334.
  • Hoff D, Zumbrun K. Multi-dimensional diffusion waves for the Navier–Stokes equations of compressible flow. Indiana Univ. Math. J. 1995;44:603–676.
  • Kobayashi T, Shibata Y. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ3. Comm. Math. Phys. 1999;200:621–660.
  • Lion P-L. Mathematics topic in fluid mechanics. Vol. 2, Compressible models, Oxford lecture series in mathematics and its applications. Oxford: Clarendon Press; 1998.
  • Mellet A, Vasseur A. On the barotropic compressible Navier–Stokes equations. Comm. Par. Differ. Equ. 2007;32:431–452.
  • Bresch D, Desjardins B. Existence of global weak solutiona for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm. Math. Phys. 2003;238:211–223.
  • Bresch D, Desjardins B, Lin C. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Par. Differ. Equ. 2003;28:843–868.
  • Haspot B. Cauchy problem for the viscous shallow water equations with a term of capillarity. Math. Models Meth. Appl. Sci. 2010;20:1049–1087.
  • Wang W, Xu C. The Cauchy problem for viscous shallow water equations. Rev. Mat. Iberoamericana. 2005;21:1–24.
  • Bresch D, Desjardins B, Métivier G. Recent mathematical results and open problems about shallow water systems, analysis and simulation of fluid dynamics. Advances in mathematical fluid mechanics. Basel: Birkhäuser; 2007. p. 15–31.
  • Chen Q, Miao C, Zhang Z. Well-posedness in critical spaces for compressible Navier–Stokes equations with density dependent viscosities. Rev. Mat. Iberoamericana. 2010;26:915–946.
  • Xin Z. Blow up of smooth solutions to the compressible Navier–Stokes equation with compact density. Comm. Pure Appl. Math. 1998;51:229–240.
  • Fujita H, Kato T. On Navier–Stokes initial value problem. Arch. Ration. Mech. Anal. 1964;16:269–315.
  • Cannone M. Ondelettes, paraproduits et Navier–Stokes, Nouveaux essais. Paris: Diderotéditeurs; 1995.
  • Cannone M. Harmonic analysis tools for solving the incompressible Navier–Stokes equations. Vol. III, Handbook of mathematical fluid dynamics. Amsterdam: North-Holland; 2004.
  • Meyer Y. Wavelets, paraproducts and Navier–Stokes equations, current developments in mathematics. Boston, MA: International Press; 1996.
  • Chemin J-Y, Lerner N. Flot de champs de vecteurs non lipschitziens et équations de Navier–Stokes [Flow of non-Lipschitz vector fields and Navier-Stokes equations]. J. Differ. Equ. 1992;121:314–328.
  • Chemin J-Y. Perfect incompressible fluids. New York (NY): Oxford University Press; 1998.
  • Chemin J-Y. Théorèmes d’unicité pour le système de Navier–Stokes tridimensionnel [Uniqueness theorems for the three-dimensional Navier-Stokes system]. J. d’Analyse Math. 1999;77:27–50.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.