Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

REGULARITY OF WEAK SOLUTIONS OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

  • Choe, Hi-Jun (Department of Mathematics Yonsei University) ;
  • Jin, Bum-Ja (Department of Mathematics Seoul National University)
  • Published : 2003.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we assume a density with integrability on the space $L^{\infty}$(0, T; $L^{q_{0}}$) for some $q_{0}$ and T > 0. Under the assumption on the density, we obtain a regularity result for the weak solutions to the compressible Navier-Stokes equations. That is, the supremum of the density is finite and the infimum of the density is positive in the domain $T^3$ ${\times}$ (0, T). Moreover, Moser type iteration scheme is developed for $L^{\infty}$ norm estimate for the velocity.

Keywords

References

  1. J. Math. Fluid Mech. v.3 no.4 On the existence of globally defined weak solutions to the Navier-Stokes equations E.Feireisl;A.Novotny;H.Petzeltova https://doi.org/10.1007/PL00000976
  2. Mathematical mechanics;Adv. Math. Fluid Mech. Recent progress in the mathematical theory of viscous compressible fluids E.Feireisl
  3. Comment. Math. Univ. Carolin v.42 no.1 On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable E.Feireisl
  4. Nonlinear analysis and its applications to differential equations; Progr. Nonlinear Differential Equations Apll On the long-time behavior of solutions to the Navier-Stokes equations of compressible flow E.Feireisl;H.Petzeltova
  5. Comm. P.D.E. v.22 Regularity of weak solutions of the compressible isentropic Navier-Stokes equations B.Desjardins https://doi.org/10.1080/03605309708821291
  6. An introduction to the Mathematical Theory of the Navier-Stokes Equations v.Ⅰ;Ⅱ G.P.Galdi
  7. Springer Tracts in Natural Philosophy v.38;39 G.P.Galdi https://doi.org/10.2996/kmj/1138846265
  8. Kodai Math. Sem. Rep. v.23 On the auchy problem for the system of fundamental equations describing movement of compressible viscous fluids N.Itaya https://doi.org/10.2996/kmj/1138846265
  9. Oxford Lecture Ser. Math. Appl v.Ⅰ;Ⅱ;10 Mathematical Topics in fluid Mechanics P.L.Lions
  10. J. Math. Pures Appl. v.77 Incompressible limit for a viscous compressible fluid P.L.Lions;N.Masmoudi https://doi.org/10.1007/BF01214738
  11. Comm. Math. Phys. v.89 Initial boundary value problems for the equations of motions of compressible viscous and heat-conductive fluids A.Matsumura;T.Nishida https://doi.org/10.1007/BF01214738
  12. J. Sov. Math. v.14 Solvability of the initial boundary value problem for the equations of a viscous compressible fluid V.A.Solonnikov https://doi.org/10.2977/prims/1195190106
  13. Publ. RIMS. Kyoto Univ. v.13 On the first initial-boundary value problem of compressible viscous fluid motion A.Tani https://doi.org/10.2977/prims/1195190106
  14. Comm. Math. Phys. v.103 Navier-Stokes equations for compressible fluids, Global existence and qualitative properties of the solutions in the general case M.Zajaczkowski;A.Valli https://doi.org/10.1007/BF01206939

Cited by

  1. A Serrin criterion for compressible nematic liquid crystal flows vol.36, pp.11, 2013, https://doi.org/10.1002/mma.2689
  2. Regularity of weak solutions of compressible isentropic self-gravitating fluid vol.147, 2016, https://doi.org/10.1016/j.na.2016.07.009
  3. A blow-up criterion for classical solutions to the compressible Navier-Stokes equations vol.53, pp.3, 2010, https://doi.org/10.1007/s11425-010-0042-6
  4. Blowup Criterion for Viscous Baratropic Flows with Vacuum States vol.301, pp.1, 2011, https://doi.org/10.1007/s00220-010-1148-y
  5. Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data vol.49, pp.1, 2017, https://doi.org/10.1137/16M1055414
  6. Serrin-Type Criterion for the Three-Dimensional Viscous Compressible Flows vol.43, pp.4, 2011, https://doi.org/10.1137/100814639
  7. Blowup criterion for viscous, compressible micropolar fluids with vacuum vol.13, pp.2, 2012, https://doi.org/10.1016/j.nonrwa.2011.08.021
  8. Blow-up criterions of strong solutions to 3D compressible Navier–Stokes equations with vacuum vol.248, 2013, https://doi.org/10.1016/j.aim.2013.07.018
  9. On the blow up criterion for the 2-D compressible Navier-Stokes equations vol.60, pp.1, 2010, https://doi.org/10.1007/s10587-010-0009-3