FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

doi:10.1016/S0252-9602(15)30082-5

Abstract

Abstract:

In this paper, we investigate the free boundary value problem(FBVP) for the cylindrically symmetric isentropic compressible Navier-Stokes equations(CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

Key words: cylindrically symmetric Navier-Stokes equations, free boundary value problem, density-dependent viscosity coefficients, strong solution

CLC Number:

35Q35

Ruxu LIAN, Jian LIU. FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J].Acta mathematica scientia,Series B, 2016, 36(1): 111-123.

Trendmd

References

[1] Gerbeau J F, Perthame B. Derivation of viscous Saint-Venant system for laminar shallow water, Numerical validation. Discrete Contin Dyn Syst Ser B, 2001,(1):89-102
[2] Marche F. Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. European J Mech B/Fluids, 2007, 26:49-63
[3] Bresch D, Desjardins B. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm Math Phys, 2003, 238:211-223
[4] Bresch D, Desjardins B. On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J Math Pures Appl, 2006, 86:362-368
[5] Fang D Y, Zhang T. Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data. J Differ Equ, 2006, 222:63-94
[6] Guo Z H, Li H L, Xin Z P. Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations. Commun Math Phys, 2012, 309:371-412
[7] Jiang S, Xin Z P, Zhang P. Global weak solutions to 1D compressible isentropy Navier-Stokes with density-dependent viscosity. Methods Appl Anal, 2005, 12:239-252
[8] Lian R X, Guo Z H, Li H L. Dynamical behaviors of vacuum states for 1D compressible Navier-Stokes equations. J Differ Equ, 2010, 248:1926-1954
[9] Liu J. Local existence of solution to free boundary value problem for compressible Navier-Stokes equations. Acta Mathematica Scientia, 2012, 32B(4):1298-1320
[10] Liu J, Lian R X. Global existence of the cylindrically symmetric strong solution to compressible Navier-Stokes equations. Abs Appl Anal, 2014, 2014:1-8
[11] Liu T P, Xin Z P, Yang T. Vacuum states for compressible flow. Discrete Contin Dynam Systems, 1998, 1998:1-32
[12] Okada M, Ne?asová Š M, Makino T. Free boundary problem for the equationof one-dimensional motion of compressible gas with density-dependent viscosity. Ann Univ Ferrara Sez VⅡ(NS), 2002, 48:1-20
[13] Vong S W, Yang T, Zhu C J. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum Ⅱ. J Differ Equ, 2003, 192:475-501
[14] Yang T, Yao Z A, Zhu C J. Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Comm Partial Differ Equ, 2001, 26:965-981
[15] Yang T, Zhao H. A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. J Differ Equ, 2002, 184:163-184
[16] Yang T, Zhu C J. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commu Math Phys, 2002, 230:329-363
[17] Mellet A, Vasseur A. Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J Math Anal, 2008, 39:1344-1365
[18] Guo Z H, Jiu Q S, Xin Z P. Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients. SIAM J Math Anal, 2008, 39:1402-1427
[19] Jiu Q S, Wang Y, Xin Z P. Stability of rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity. Comm Partial Differ Equ, 2011, 36:602-634
[20] Jiu Q S, Xin Z P. The Cauchy problem for 1D compressible flows with density-dependent viscosity coeffi-cients. Kinet Relat Modeks, 2008, 1:313-330
[21] Li H L, Li J, Xin Z P. Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Commu Math Phys, 2008, 281:401-444
[22] Lian R X, Liu J, Li H L, Xiao L. Cauchy problem for the one-dimensional compressible Navier-Stokes equations. Acta Mathematica Scientia, 2012, 32B(1):315-324
[23] Solonnikov V A, Tani A. Free Boundary Problem for a Viscous Compressible Flow with a Surface Tension//Constantin Carathéodory:an International Tribute, Vol I, Ⅱ. Teaneck, NJ:World Sci Publ, 1991:1270-1303
[24] Zajaczkowski W M. Existence of local solutions for free boundary problems for viscous compressible barotropic fluids. Ann Polon Math, 1995, 60:255-287
[25] Zajaczkowski W M. On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface. Dissert Math, 1993, 324:1-101
[26] Secchi P, Valli A. A free boundary problem for compressible viscous fluids. J Reine Angew Math, 1983, 341:1-31
[27] Tani A. On the free boundary value problem for compressible viscous fluid motion. J Math Kyoto Univ, 1981, 21:839-859
[28] Zadrzyńska E, Zajaczkowski W M. On local motion of a general compressible viscous heat conducting fluid bounded by a free surface. Ann Polon Math, 1994, 59:133-170
[29] Solonnikov V A, Tani A. Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid//The Navier-Stokes Equations Ⅱ-Theory and Numerical Methods(Oberwolfach, 1991). Lecture Notes in Math, 1530. Berlin:Springer, 1992:30-55
[30] Zajaczkowski W M. On nonstationary motion of a compressible barotropic viscous capillary fluid bounded by a free surface. SIAM J Math Anal, 1994, 25:1-84
[31] Zadrzyńska E. Evolution free boundary problem for equations of viscous compressible heat-conducting capillary fluids. Math Meths Appl Sci, 2001, 24:713-743
[32] Zadrzyńska E, Zajaczkowski W M. On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting capillary fluid. J Appl Anal, 1996, 2:125-169
[33] Frid H, Shelukhin V V. Boundary layer for the Navier-Stokes equations of compressible fluids. Commun Math Phys, 1999, 208:309-330
[34] Frid H, Shelukhin V V. Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry. SIAM J Math Anal, 2000, 31:1144-1156
[35] Fan J S, Jiang S. Zero shear viscosity limit for the Navier-Stokes equations of compressible isentropic fluids with cylindric symmetry. Rend Sem Mat Univ Politec Torino, 2007, 65:35-52
[36] Jiang S, Zhang J W. Boundary layers for the Navier-Stokes equations of compressible heat-conducting fiows with cylindrical symmetry. SIAM J Math Anal, 2009, 41:237-268
[37] Yao L, Zhang T, Zhu C J. Boundary layers for compressible Navier-Stokes equations with density-dependent viscosity and cylindrical symmetry. Ann Inst H Poincaré Anal Non Linéaire, 2011, 28:677-709