带粘性的波动方程组解的逐点估计

Abstract

Abstract:

The Cauchy problem for two systems of wave equations with viscosity in dimension three is considered. By using the long wave and short wave decomposition method together with energy method and Green function, the pointwise estimates of the time-asymptotic shape of the solution are given, which exhibit two kinds of generalized Huygens' waves. As a byproduct, the optimal L^p-decay rates with p ≥ 1 of the solutions of these systems are also established.

Key words: Green function, Systems of wave equations with viscosity, Huygens' principle

CLC Number:

O175.2

Zhigang Wu,Xiaofang Miao. Pointwise Estimates for Systems of Wave Equations with Viscosity[J].Acta mathematica scientia,Series A, 2019, 39(6): 1421-1442.

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References 27

1	Chen G W , Wang Y P . A note on the existence of solutions of quasilinear wave equations with viscosity. Nonlinear Anal TMA, 2008, 68: 609- 620 doi: 10.1016/j.na.2005.04.057
2	Deng S J , Wang W K , Zhao H L . Existence theory and L^p estimates for the solution of nonlinear dissipative wave equation. Nonlinear Anal RWA, 2010, 11: 4404- 4414 doi: 10.1016/j.nonrwa.2010.05.024
3	Diasa J P , Fridb H . Radially symmetric weak solutions for a quasilinear wave equation in two space dimensions. J Diff Eqns, 2005, 219: 306- 322 doi: 10.1016/j.jde.2005.02.004
4	Hosono T , Ogawa T . Large time behavior and L^p-L^q estimate of solutions of 2-dimensional nonlinear damped wave equations. J Diff Eqns, 2004, 203: 82- 118 doi: 10.1016/j.jde.2004.03.034
5	Kobayashi T , Pecher H , Shibata Y . On the global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity. Math Ann, 1993, 296: 215- 234 doi: 10.1007/BF01445103
6	Liao J , Wang W K , Yang T . L^p convergence rates of planar waves for multi-dimensional Euler equations with damping. J Diff Eqns, 2009, 247: 303- 329 doi: 10.1016/j.jde.2009.03.011
7	Li T T , Chen Y M . Nonlinear Evolution Function (in Chinese). Beijing:Science Press, 1989
8	Liu T P , Wang W K . The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. Comm Math Phys, 1998, 196: 145- 173 doi: 10.1007/s002200050418
9	Liu T P , Noh S E . Wave propagation for the compressible Navier-Stokes equations. J Hyperbolic Differ Eqns, 2015, 12: 385- 445 doi: 10.1142/S0219891615500113
10	Liu Y Q , Wang W K . The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete Contin Dyn Syst, 2008, 20: 1013- 1028 doi: 10.3934/dcds.2008.20.1013
11	Liu Y Q . The pointwise estimates of solutions for semilinear dissipative wave equation. Comm Pure Appl Anal, 2013, 12: 237- 252
12	Marchti P , Nishihara K . The L^p-L^q estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media. J Diff Eqns, 2003, 191: 445- 469 doi: 10.1016/S0022-0396(03)00026-3
13	Nakao M . Energy decay for the quasilinear wave equation with viscosity. Math Z, 1995, 219: 289- 299 doi: 10.1007/BF02572366
14	Nakao M . Existence of an anti-periodic solution for the quasilinear wave equation with viscosity. J Math Anal Appl, 1996, 204: 754- 764
15	Nishihara K . L^p-L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application. Math Z, 2003, 244: 631- 649 doi: 10.1007/s00209-003-0516-0
16	Ono K . L^p-L^q decay problem for the dissipative wave equation in odd dimensions. J Math Anal Appl, 2005, 310: 347- 361 doi: 10.1016/j.jmaa.2004.12.057
17	Ono K . L^p decay problem for the dissipative wave equation in even dimensions. Math Methods Appl Sci, 2004, 27: 1843- 1863 doi: 10.1002/mma.523
18	Shibata Y . On the rate of decay of solutions to linear viscoelastic equation. Math Methods Appl Sci, 2000, 23: 203- 226 doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M
19	Wang W K , Yang T . The pointwise estimates of solutions for Euler equations with damping in multi-dimensions. J Diff Eqns, 2001, 173: 410- 450 doi: 10.1006/jdeq.2000.3937
20	Wang W K , Wu Z G . Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions. J Diff Eqns, 2010, 248: 1617- 1636 doi: 10.1016/j.jde.2010.01.003
21	Wang W J . Pointwise estimates for solutions to a system of nonlinear damped wave equations. Electronic J Diff Eqns, 2013, 113: 1- 16
22	Wu Z G , Wang W K . Generalized Huygens' principle for a reduced gravity two and a half layer model in dimension three. Kinetic Related Models, 2017, 10: 1205- 1233 doi: 10.3934/krm.2017046
23	Wu Z G , Wang W K . Pointwise Estimates for Bipolar Compressible Navier-Stokes-Poisson System in Dimension Three. Archive Rational Mechanics Analysis, 2017, 226: 587- 638 doi: 10.1007/s00205-017-1140-1
24	Wu Z G , Wang W K . Pointwise estimates of solution for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions. Acta Math Sci, 2012, 32: 1681- 1702 doi: 10.1016/S0252-9602(12)60134-9
25	Wu Z G , Wang W K . Refined pointwise estimates for the Navier-Stokes-Poisson equations. Analysis and Applications, 2016, 14: 739- 762 doi: 10.1142/S0219530515500153
26	Xu H M , Wang W K . Pointwise estimates for the wave equation with dissipation in odd spatial dimension. J Hyperbolic Diff Eqns, 2008, 5: 477- 486 doi: 10.1142/S0219891608001556
27	Xu H M , Wang W K . Large time behavior of solutions for the nonlinear wave equation with viscosity in odd dimensions. Nonlinear Anal TMA, 2012, 75: 543- 549 doi: 10.1016/j.na.2011.08.056

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Pointwise Estimates for Systems of Wave Equations with Viscosity

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