STABILITY OF SHOCK PROFILES FOR NONCONVEX SCALAR VISCOUS CONSERVATION LAWS

This paper is to study the stability of shock profiles for nonconvex scalar viscous conservation laws with the nondegenerate and the degenerate shock conditions by means of an elementary energy method. In both cases, the shock profiles are proved to be asymptotically stable for suitably small initial disturbances. Moreover, in the case of nondegenerate shock condition, time decay rates of asymptotics are also obtained.

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Optimal time‐decay rates for the 3D compressible Magnetohydrodynamic flows with discontinuous initial data and large oscillations

Journal of the London Mathematical Society ◽

10.1112/jlms.12393 ◽

2020 ◽

Author(s):

Guochun Wu ◽

Yinghui Zhang ◽

Weiyuan Zou

Keyword(s):

Initial Data ◽

Decay Rates ◽

Optimal Time ◽

Time Decay ◽

Discontinuous Initial Data ◽

Magnetohydrodynamic Flows ◽

Time Decay Rates

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Remarks on the stability of shock profiles for conservation laws with dissipation

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1985-0797065-0 ◽

1985 ◽

Vol 291 (1) ◽

pp. 353-353 ◽

Cited By ~ 14

Author(s):

Robert L. Pego

Keyword(s):

Conservation Laws ◽

Shock Profiles ◽

The Stability

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Sharp time decay rates of H 1 weak solutions for the 2D MHD equations with linear damping velocity

Nonlinearity ◽

10.1088/1361-6544/ab8f7d ◽

2020 ◽

Vol 33 (9) ◽

pp. 4857-4877

Author(s):

Hailong Ye ◽

Yan Jia ◽

Bo-Qing Dong

Keyword(s):

Weak Solutions ◽

Decay Rates ◽

Mhd Equations ◽

Time Decay ◽

Time Decay Rates ◽

Linear Damping

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Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

Science China Mathematics ◽

10.1007/s11425-020-1719-9 ◽

2020 ◽

Author(s):

Guochun Wu ◽

Yinghui Zhang ◽

Anzhen Zhang

Keyword(s):

Global Existence ◽

Decay Rates ◽

Navier Stokes ◽

Time Decay ◽

Poisson System ◽

Time Decay Rates

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A note on the -stability of boundary layers in viscous conservation laws

Nonlinear Analysis ◽

10.1016/j.na.2007.02.014 ◽

2008 ◽

Vol 68 (9) ◽

pp. 2677-2680 ◽

Cited By ~ 1

Author(s):

R. Cavazzoni

Keyword(s):

Conservation Laws ◽

Boundary Layers ◽

Viscous Conservation Laws ◽

The Stability

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The Boltzmann equation, Besov spaces, and optimal time decay rates inRxn

Advances in Mathematics ◽

10.1016/j.aim.2014.04.012 ◽

2014 ◽

Vol 261 ◽

pp. 274-332 ◽

Cited By ~ 44

Author(s):

Vedran Sohinger ◽

Robert M. Strain

Keyword(s):

Boltzmann Equation ◽

Besov Spaces ◽

Decay Rates ◽

Optimal Time ◽

Time Decay ◽

The Boltzmann Equation ◽

Time Decay Rates

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Time decay rates for the compressible viscoelastic flows

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2017.03.044 ◽

2017 ◽

Vol 452 (2) ◽

pp. 990-1004 ◽

Cited By ~ 3

Author(s):

Guochun Wu ◽

Zhensheng Gao ◽

Zhong Tan

Keyword(s):

Decay Rates ◽

Viscoelastic Flows ◽

Time Decay ◽

Time Decay Rates

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Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

Journal of Differential Equations ◽

10.1016/j.jde.2007.10.022 ◽

2008 ◽

Vol 244 (1) ◽

pp. 87-116 ◽

Cited By ~ 2

Author(s):

Margaret Beck ◽

C. Eugene Wayne

Keyword(s):

Conservation Laws ◽

Traveling Waves ◽

Invariant Manifolds ◽

Viscous Conservation Laws ◽

The Stability

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Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates

Journal of Evolution Equations ◽

10.1007/s00028-018-0426-6 ◽

2018 ◽

Vol 18 (2) ◽

pp. 923-946 ◽

Cited By ~ 1

Author(s):

Franz Achleitner ◽

Yoshihiro Ueda

Keyword(s):

Asymptotic Stability ◽

Conservation Laws ◽

Traveling Wave ◽

Traveling Wave Solutions ◽

Decay Rates ◽

Wave Solutions ◽

Viscous Conservation Laws

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Lp ENERGY METHOD FOR MULTI-DIMENSIONAL VISCOUS CONSERVATION LAWS AND APPLICATION TO THE STABILITY OF PLANAR WAVES

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891604000196 ◽

2004 ◽

Vol 01 (03) ◽

pp. 581-603 ◽

Cited By ~ 29

Author(s):

SHUICHI KAWASHIMA ◽

SHINYA NISHIBATA ◽

MASATAKA NISHIKAWA

Keyword(s):

Conservation Laws ◽

Energy Method ◽

Optimal Convergence Rate ◽

Estimates Of Solutions ◽

Interpolation Inequalities ◽

Optimal Decay ◽

Viscous Conservation Laws ◽

The Cauchy Problem ◽

Whole Space ◽

The Stability

We introduce a new Lp energy method for multi-dimensional viscous conservation laws. Our energy method is useful enough to derive the optimal decay estimates of solutions in the W1,p space for the Cauchy problem. It is also applicable to the problem for the stability of planar waves in the whole space or in the half space, and gives the optimal convergence rate toward the planar waves as time goes to infinity. This energy method makes use of several special interpolation inequalities.

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