Decay Estimates Of Solutions For The P-system With General Linear Dissipation

In the Lagrangian coordinates, isentropic ideal fluid mechanics equations in one dimension canbe rewritten into the p-system, it’s one of the hot issues. This paper mainly deals with the decayproperty of solutions for the p-system with general linear dissipation. So far, there are manyresearches about the similar problem. Wang Weike and Yang Tong had been studied the decayproperty of the solutions for the p-system with special damping and obtained the L~p1(＜p≤∞) convergence rate of the solution in multi-dimensions. Compared with their results, this paper isinterested in more general situation and from the decay estimates of the solutions we can be intuitiveto see the hyperbolic behavior of the equations. The main methods this paper used are the Fourieranalysis method and the Green function method in combination with the fundamental solution used inthe Cauchy problem for partial differential equations. Under some suitable conditions on the initialdata and the dissipative coefficients, this article gives a detailed description for the decay property ofthe solutions for the p-system with general linear dissipation. In addition, the discussion can beextended to more general2×2hyperbolic equations with the same kind of dissipation.This paper is divided into five chapters altogether, and the main contents and results are listed asfollows:Firstly, we use the Fourier analysis method give a detailed description for the decay property ofthe solutions for the general2×2hyperbolic equations with general linear dissipation.Secondly, the condition can be weakened and it can also get the decay property of solutions forthe general2×2hyperbolic equations.Finally, based on the similar analysis of the p-system with general linear dissipation, theequations fulfill the weakend conditions, and as a result it can get the decay property and the decayestimates of solutions for the p-system with general linear dissipation.