Acta mathematica scientia, Series B >
NONLINEAR EVOLUTION SYSTEMS AND GREEN'S FUNCTION
Received date: 2010-09-06
Online published: 2010-11-20
Supported by
The authors are supported by National Science Foundation of China (11071162) and Shanghai Municipal Natural Science Foundation (09ZR1413500).
In this paper, we will introduce how to apply Green's function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen's principle. Then, for other nonlinear
dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green's function of the linearized system and micro-local analysis, such as frequency decomposition and so on.
WANG Wei-Ke . NONLINEAR EVOLUTION SYSTEMS AND GREEN'S FUNCTION[J]. Acta mathematica scientia, Series B, 2010 , 30(6) : 2051 -2063 . DOI: 10.1016/S0252-9602(10)60190-7
[1] Caffarelli L, Vasseur, A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann Math, 2010, 171(3): 1903--1930
[2] Cazenave T. Semilinear Schr\"{o}dinger Equation. Courant Lecture Notes in Math, 10. Courant Ins Math Sci and Amer Math Soc, 2003
[3] Constantin P, Wu J H. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J Math Anal, 1999, 30(5): 937--948 (electronic)
[4] Cordoba D. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann Math, 1998, 148: 1135--1152
[5] Cordoba A, Cordoba D. A maximum principle applied to quasi-geostrophic equation. Comm Math Phys, 2004, 249: 511--528
[6] Deng S J, Wang W K. Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation. to appear in Discrete and Continuous Dynamical Systems (A)
[7] Deng S J, Wang W K, Yu S H. Pointwise convergence to Knudsen layers of the Boltzmann equation. Comm Math Phys, 2008, 281: 287--347
[8] Evans L C. Partial Differential Equations. Graduate Studies in Math, Vol 19. Amer Math Soc, 1998
[9] Gao W L, Zhu C J. Asymptotic decay toward the planar rarefaction waves for a model system of the radiating gas in two dimensions. Math Models Methods Appl Sci, 2008, 18: 511--541
[10] Hammer K. Nonlinear effects on the propagation of sound waves in a radiating gas. Quart J Mech Appl Math, 1971, 24: 155--168
[11] H\"{o}rmander L. Lecture on Nonlinear Hyperbolic Differential Equations. Mathematiques \& Application 26. Springer-Verlag, 1997
[12] Kawashima S. System of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics
[D]. Kyoto: Kyoto University, 1983
[13] Kiselev A, Nazarov F, Volberg A. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent Math, 2007, 167(3): 445--453
[14] Kreiss H O, Lorens J. Initial-boundary value problems and the Navier-Stokes equations. Appl Math 47. SIAM, 2004
[15] Li T T, Chen Y M. Nonlinear Evolution Equations. Beijing: Science Press, 1989 (in Chinese)
[16] Liu, T P. Pointwise convergence to shock waves for viscous conservation laws. Comm Pure Appl Math, 1997, 50(11): 1113--1182
[17] Liu T P, Wang W K. The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. Comm Math Phy,
1998, 196: 145--173
[18] Liu T P, Yu S H. Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation. Comm Pure Appl Math, 2007, 60: 295--356
[19] Liu T P, Zeng Y. Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws. Amer Math Soc Memoirs, 1997, 125(599)
[20] Markowich P A, Ringhofer C, Schmeiser C. Semiconductors Equations. Vienna, New York: Springer, 1990
[21] Metivier G. Small Viscosity and Boundary Layer Methods. Birkhauser, 2003
[22] Shizuta Y, Kawashima S. Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math J, 1985, 14(2): 249--275
[23] Ukai S, Yang T. Mathematical Theory of Boltzmann Equation. Lecture Notes in City University of Hong Kong, 2006
[24] Vincenti W G, Kruger C H. Introduction to Physical Gas Dynamics. New York: Wiley & Sons, 1965
[25] Wang W K. The pointwise estimates of solutions for general Navier-Stokes systems in odd multi-dimensions. Methods Appl Anal, 2005, 12(3): 279--290
[26] Wang W K. The pointwise estimate of solutions for Navier-Stokes equations in multi-dimensions//Hyperbolic Problems Theory, Numerics and Applications (Tenth international conference in Osaka). Yokohama Publishers, 2006: 205--212
[27] Wang W K, Wang W J. The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions. Nonlinear Anal, 2009, 71: 1180--1195
[28] Wang W K, Wang W J. Global existence of solutions for a model system of the radiating gas with large initial data. Preprint
[29] Wang W K, Wu Z G. Pointwise estimate of solutions for the Navier-Stokes-Poisson equation in multi-dimensions. J Differ Equ, 2010, 248: 1617--1636
[30] Wang W K, Xu H M. Pointwise estimate of solutions of isentropic Navier-Stokes equations in even multi-dimensions. Acta Math Sci, 2001, 21B(3): 417--427
[31] Wang W K, Yang T. The pointwise estimates of solutions for Euler equations with daping in multi-dimensions. J Differ Equ, 2001, 173: 410--450
[32] Wang W K, Yang, T. Existence and stability of planar diffusion waves for 2-D Euler equations with damping. J Differ Equ, 2007, 242(1): 40--71
[33] Wang W K, Yang X F. The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions. J Hyperbolic Differ Equ, 2005, 2(3): 673--695
[34] Wu Z G, Wang W K. Pointwise estimate of solutions for the Euler-Poisson equation with damping in multi-dimensions. Discrete and Continuous Dynamical Systems (A), 2010, 26: 1101--1117
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