LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

数学物理学报 ›› 2014, Vol. 34 ›› Issue (3): 851-871.

LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

叶嵎林|窦昌胜|酒全森

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China； School of Statistics, Capital University of Economics and Business, Beijing 100070, China；LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

收稿日期:2013-06-28 修回日期:2013-09-09 出版日期:2014-05-20 发布日期:2014-05-20
基金资助:
The research is partially supported by China Post-doctoral Science Foundation (2012M520205); the research is partially supported by National Natural Sciences Foundation of China (11171229, 11231006) and Project of Beijing Chang Cheng Xue Zhe.

LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

YE Yu-Lin, DOU Chang-Sheng, JIU Quan-Sen

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China； School of Statistics, Capital University of Economics and Business, Beijing 100070, China；LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received:2013-06-28 Revised:2013-09-09 Online:2014-05-20 Published:2014-05-20
Supported by:
The research is partially supported by China Post-doctoral Science Foundation (2012M520205); the research is partially supported by National Natural Sciences Foundation of China (11171229, 11231006) and Project of Beijing Chang Cheng Xue Zhe.

摘要/Abstract

摘要：

In this article, we prove the local existence and uniqueness of the classical solution to the Cauchy problem of the 3-D compressible Navier-Stokes equations with large initial data and vacuum, if the shear viscosity μ is a positive constant and the bulk viscosity λ(ρ) = ρ^βwith β ≥0. Note that the initial data can be arbitrarily large to contain vacuum states.

关键词: Existence and uniqueness, classical solution, compressible Navier-Stokes equa-tions, density-dependent viscosity, vacuum

Abstract:

Key words: Existence and uniqueness, classical solution, compressible Navier-Stokes equa-tions, density-dependent viscosity, vacuum

中图分类号:

35A09

叶嵎林, 窦昌胜, 酒全森. LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报, 2014, 34(3): 851-871.

YE Yu-Lin, DOU Chang-Sheng, JIU Quan-Sen. LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. Acta mathematica scientia,Series A, 2014, 34(3): 851-871.

参考文献

[1] 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(1/2): 211–223

[2] Bresch D, Desjardins B. Quelques modeles diffusifs capillaires de type Korteweg. C R Acad Sci, Paris, section mecanique, 2004, 332 (11): 881–886

[3] 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

[4] Bresch D, Desjardins B, Lin Chi-Kun. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm Partial Differential Equations, 2003, 28(3/4): 843–868

[5] Bresch D, Desjardins B, Gerard-Varet D. On compressible Navier-Stokes equations with density dependent
viscosities in bounded domains. J Math Pures Appl, 2007, 87(2): 227–235

[6] Cho Y, Kim H. Existence results for viscous polytropic fluids with vacuum. Hokkaido University Preprint Series in Math, 2003, 675

[7] Cho Y, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluids. J Math Pures Apple, 2004, 83: 243–275

[8] Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equation with nonnegative initial densities. Manuscripta Math, 2006: 91–129

[9] Danchin R. Global existence in critical spaces for compressible Navier-Stokes equations. Invent Math, 2000, 141: 579–614

[10] Dou C S, Jiu Q S. A remark on free boundary problem of 1-D compressible Navier-Stokes equations with density-dependent viscosity. Math Meth Appl Sci, 2010, 33: 103–116

[11] Feireisl E, Novotn´y A, Petzeltov´a H. On the existence of globally defined weak solutions to the Navier-Stokes
equations of isentropic compressible fluids. J Math Fluid Mech, 2001, 3: 358–392

[12] Galdi G P. An introduction to the mathematical theory of the Navier-Stokes equations. New York: Springer-Verlag, 1994

[13] Gerbeau J F, Perthame B. Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numer-ical Validation. Discrete and Continuous Dynamical Systems, 2001, 1B(1): 89–102

[14] 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(5): 1402–1427

[15] Guo Z H, Li H L, Xin Z P. Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations. (English) Commun Math Phys, 2012, 309(2): 371–412

[16] Hoff D. Global existence of 1D compressible isentropic Navier-Stokes equations with large initial data. Trans Amer Math Soc, 1987, 303(1): 169–181

[17] Hoff D. Compressible flow in a half-space with Navier boundary conditions. J Math Fluid Mech, 2005, 7(3): 315–338

[18] Hoff D. Strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data. Arch Rat Mech Anal, 1995, 132: 1–14

[19] Hoff D. The zero-Mach limit of compressible flows. Comm Math Phys, 1998, 192: 543–554

[20] Huang X D, Li J. Global well-posedness of classical solutions to the Cauchy problem of two-dimesional baratropic compressible Navier-Stokes system with vacuum and large initial data. preprint. arxiv:1207.3746

[21] Huang X D, Li J. Existence and blowup behavior of global strong solutions to the two-dimensional baratropic
compressible Navier-Stokes system with vacuum and large initial data. http://arxiv.org/abs/1205.5342

[22] Hoff D, Serre D. The failure of continuous dependence on initial data for the Navier-Stokes equations of
compressible flow. SIAM J Appl Math, 1991, 51: 887–898

[23] Hoff D, Smoller J. Non-formation of vacuum states for compressible Navier-Stokes equations. Comm Math Phys, 2001, 216(2): 255–276

[24] Itaya N. On the Cauchy problem for the system of fundamental quations desctibing the movement of compressible viscous fluids. Kodai Math Sep, 1971, 23: 60–120

[25] Jiang S. Global smooth solutions of the equations of a viscous, heat-conducting one-dimensional gas with density-dependent viscosity. Math Nachr, 1998, 190: 169–183

[26] Jiang S, Xin Z P, Zhang P. Global weak solutions to 1D compressible isentropy Navier-Stokes with density-
dependent viscosity. Methods and Applications of Analysis, 2005, 12(3): 239–252

[27] Jiu Q S, Xin Z P. The Cauchy problem for 1D compressible flows with density-dependent viscosity coeffi-cients. Kinet Relat Models, 2008, 1(2): 313–330

[28] 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 Differential Equations, 2011, 36: 602–634

[29] Jiu Q S, Wang Y, Xin Z P. Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum. arXiv:1202.1382v1

[30] Jiu Q S, Wang Y, Xin Z P. Global well-posedness of the Cauchy problem of 2D compressible Navier-Stokes
equations in weighted spaces. J Dierential Equations, 2013, 255(3): 351–404

[31] Jiu Q S, Wang Y, Xin Z P. Global classical solutions to the two-dimensional compressible Navier-Stokes equations in R2. http://arxiv.org/abs/1209.0157vl

[32] Jurgen J. Partial Differential Equations. Second Edition. New York: Springer-Verlag, 2006: 224

[33] Kanel J I. A model system of equations for the one-dimensional motion of a gas. (Russian) Differencial’nye
Uravnenija, 1968, 4: 721–734

[34] Kazhikhov A V, Shelukhin V V. Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J Appl Math Mech, 1977, 41: 273–282; Translated from: Prikl Mat Meh, 1977, 41 : 282–291

[35] Li H L, Li J, Xin Z P. Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations. Comm Math Phys, 2008, 281(2): 401–444

[36] Lions P L. Existence globale de solutions pour les equations de Navier-Stokes compressibles isentropoques.
C R Acad Sci, 1993, 316: 1335–1340

[37] Lions P L. Mathematical Topics in Fluid Dynamics 2, Compressible Models. Oxford: Oxford Science Publication, 1998

[38] Liu T P, Xin Z P, Yang T. Vacuum states of compressible flow. Discrete Continuous Dynam systems, 1998, 4(1): 1–32

[39] 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(1): 67–104

[40] Matsumura A, Nishida T. The initial boundaty value problems for the equations of motion of compressible and heat-conductive fluids. Comm Math Phys, 1983, 89: 445–464

[41] Mellet A, Vasseur A. On the isentropic compressible Navier-Stokes equation. Comm Partial Differential Equations, 2007, 32: 431–452

[42] Mellet A, Vasseur A. Existence and uniqueness of global strong solutions for one-dimensional compressible
Navier-Stokes equations. SIAM J Math Anal, 2008, 39(4): 1344–1365

[43] Nash J. Le problem de Cauchy pour les equations differentielles d’un fluide general. Bull Soc Math France, 1962, 90: 487–497

[44] Okada M, Matusu-Necasova S, Makino T. Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity. Ann Univ Ferrara Sez VII (NS), 2002, 48: 1–20

[45] Perepelitsa M. On the global existence of weak solutions for the Navier-Stokes equations of compressible fluid flows. SIAM J Math Anal, 2006, 38(1): 1126–1153

[46] Serre D. Sur l´equation monodimensionnelle d´un fluide visqueux, compressible et conducteur de chaleur. C R Acad Sci Paris S´er I, 1986, 303: 703–706

[47] Vaigant V A, Kazhikhov A V. On existence of global solutions to the two-dimensional Navier-Stokes equa-tions for a compressible viscous fluid. Siberian J Math, 1995, 36: 1283–1316

[48] Tani A. On the first initial-boundary value problem of compressible viscous fluid motion. Publ Res Inst Math Sci Kyoto Univ, 1971, 13: 193–253

[49] Vong S W, Yang T, Zhu C J. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum II. J Differential Equations, 2003, 192(2): 475–501

[50] Xin Z P. Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density. Comm Pure Appl Math, 1998, 51: 229–240

[51] Yang T, Yao Z A, Zhu C J. Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Comm Partial Differential Equations, 2001, 26(5/6): 965–981

[53] Yang T, Zhao H J. A vacuum problem for the one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity. J Differential Equations, 2002, 184(1): 163–184

[53] Yang T, Zhu C J. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Comm Math Phys, 2002, 230(2): 329–363

[54] Zhang T, Fang D Y. Existence and uniqueness results for viscous, heat-conducting 3-D fluid with vacuum. J Partial Differential Equations, 2008, 21(4): 347–376

[55] Zhang T, Fang D Y. Compressible flows with a density-dependent viscosity coefficient. SIAM J Math Anal, 2009/2010, 41(6): 2453–2488

LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

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