Large Time Behavior Of The Solution To The Initial-boundary Value Problem Of Some Hyperbolic Equations With Dissipative Terms

This dissertation mainly study the large time behavior of the solution to aninitial boundary value problem of there hyperbolic equations with dissipative terms.The first problem is the stability and decay problem of two-dimensional viscous con-servation laws, the second is the initial-boundary value problem of Navier-Stokesequations with a supersonic boundary condition and the last one is the existenceand decay problem of the solutions to initial-boundary value problem of the con-servation laws with relaxation terms. For these hyperbolic equations, by addingboundary value conditions to make them well-posed, we study their large time be-havior of solutions via weighted energy method. In the study of initial-boundaryvalue problem for Navier-Stokes equations, H. Keiss and J. Lorenz have proposedthat how to give boundary value condition to hyperbolic equations depends onthe sign of the equation’s eigenvalue, the number of boundary value conditionsneeded to make the problem well-posed should agree with the number of positiveeigenvalues of equations.As we know, in the study of conservation laws, it is very important to considerthe conservation laws with dissipative structure. There are three diferent kinds ofmechanisms of the usual conservation laws with dissipative terms. they are con-servation laws with viscous, with damping terms and with relaxation respectively.Although they are all dissipative mechanisms, their impacts on the systems aretotally diferent. As well known, the main part of the solution to Navier-Stokesequations moves along the cone x=ct, but the main part of the solution to con- servation laws with relaxation travels along a certain direction determined by therelaxation mechanism in steady of the characteristic cone of the hyperbolic part.The precondition to get the exponential decay of solutions is to give the bound-ary value condition at right positions: based on the wave propagation property, wegive the proper boundary value condition to make the wave move to the outsideof the considered region so that the main part of the solution exists outside of theboundary, which show the exponential decay is possible and via the exponentialweighted energy method we did get the exponential decay of solutions.In this dissertation, we get the pointwise of solution by the weighted energymethod.Concretely, we have done the following works.Chapter one is an introduction, we introduce the research backgrounds, pro-gresses about this topic and main works of this paper, some preliminary theoriesand approaches.In Chapter two, we study the initial-boundary value problem of viscous con-servation laws. For the non-degenerate case f (u+) b <0at spatial infinity, weinvestigate the convergence rate of solution toward the boundary layer solution.Precisely, we obtain an algebraic and exponential decay rate if the initial perturba-tion decay is the algebraic and exponential decay rate. These results are proved bythe estimation in the H2Sobolev space and the weighted energy method. Theseresults do not need the smallness of initial data.In Chapter three, we study the initial-boundary value problem of Navier-Stokes equations with a supersonic boundary. The main part of the solution movetoward to outside of the boundary. Given a constant equilibrium state (ρ,0), weconstruct the global existence of solutions. By using weighted energy estimates,we show that the solution converges to the equilibrium state with exponential ratewhen the perturbations are sufciently small.In Chapter four, we study the initial-boundary value problem of conservation laws with relaxation. The nonlinear part can be closed based on a priori assumptionand the estimates for derivatives from energy method. The global existence canbe obtained based on local existence theorem. Via the weighted energy method,we get the pointwise estimates for the solutions.