Asymptotic stability of planar rarefaction waves for scalar viscous conservation laws in several dimensions

1989 ◽  
pp. 299-303
Author(s):  
Zhou Ping Xin
Keyword(s):  
Asymptotic Stability ◽  
Conservation Laws ◽  
Rarefaction Waves ◽  
Viscous Conservation Laws

Degenerate boundary layers for a system of viscous conservation laws

2016 ◽  
Vol 14 (01) ◽  
pp. 75-99
Author(s):  
Tohru Nakamura
Keyword(s):  
Boundary Layer ◽  
Asymptotic Stability ◽  
Conservation Laws ◽  
Type Inequality ◽  
A Priori Estimate ◽  
Hardy Type Inequality ◽  
Weighted Energy Method ◽  
Boundary Layer Solution ◽  
Viscous Conservation Laws ◽  
Layer Solution

This paper is concerned with existence and asymptotic stability of a boundary layer solution which is a smooth stationary wave for a system of viscous conservation laws in one-dimensional half space. With the aid of the center manifold theory, it is shown that the degenerate boundary layer solution exists under the situation that one characteristic is zero and the other characteristics are negative. Asymptotic stability of the degenerate boundary layer solution is also proved in an algebraically weighted Sobolev space provided that the weight exponent [Formula: see text] satisfies [Formula: see text]. The stability analysis is based on deriving the a priori estimate by using the weighted energy method combined with the Hardy type inequality with the best possible constant.

ASYMPTOTIC DECAY TOWARD THE PLANAR RAREFACTION WAVES OF SOLUTIONS FOR VISCOUS CONSERVATION LAWS IN SEVERAL SPACE DIMENSIONS

1996 ◽  
Vol 06 (03) ◽  
pp. 315-338 ◽  
Author(s):  
KAZUO ITO
Keyword(s):  
Conservation Laws ◽  
Decay Rate ◽  
Energy Method ◽  
Rarefaction Waves ◽  
Asymptotic Decay ◽  
One Dimensional ◽  
Viscous Conservation Laws ◽  
Asymptotic Decay Rate

This paper gives the asymptotic decay rate toward the planar rarefaction waves of the solutions for the scalar viscous conservation laws in two or more space dimensions. This is proved by a result on the decay rate of solutions for one-dimensional scalar viscous conservation laws and by using an L2-energy method with a weight of time.

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