Shock reflection for a system of hyperbolic balance laws

This paper concerns shock reflection for a system of hyperbolic balance laws in one space dimension. It is shown that the generalized nonlinear initial-boundary Riemann problem for a system of hyperbolic balance laws with nonlinear boundary conditions in the half space admits a unique global piecewise C¹ solution u=u(t,x) containing only shocks with small amplitude and this solution possesses a global structure similar to that of self-similar solution of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shocks but no centered rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory.     

Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities

In this paper, the author proves the global structure stability of the Lax's Riemann solution , containing only shocks and contact discontinuities, of general n×n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C¹ solution u=u(t,x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws

This work is a continuation of our previous work, in the present paper we study the generalized nonlinear initial-boundary Riemann problem with small BV data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space . We prove the global existence and uniqueness of piecewise C¹ solution containing only contact discontinuities to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem, for general n×n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution to the corresponding nonlinear initial-boundary Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given.     

Global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary

This work is a continuation of our previous work, in the present paper we study the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws with non-linear boundary conditions in the half space . Under the assumption that each characteristic with positive velocity is linearly degenerate, we prove the existence and uniqueness of global weakly discontinuous solution u=u(t,x) with small amplitude, and this solution possesses a global structure similar to that of the self-similar solution of the corresponding Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines, particularly to the system describing the motion of the relativistic string in Minkowski space R1+n, are also given.     

Exponential stability of wave equations with potential and indefinite damping

First, we consider the linear wave equation utt−uxx+a(x)ut+b(x)u=0 on a bounded interval (0,L)⊂R. The damping function a is allowed to change its sign. If is positive and the spectrum of the operator (_∂xx−b) is negative, exponential stability is proved for small . Explicit estimates of the decay rate ω are given in terms of and the largest eigenvalue of (_∂xx−b). Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equation utt−σx(ux)+a(x)ut+b(x)u=0, if, additionally, the negative part of a is small enough compared with ω.     

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies as t→∞, in , where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.     

Localization for the evolution p-Laplacian equation with strongly nonlinear source term

In this paper we study the strict localization for the p-Laplacian equation with strongly nonlinear source term. Let u:=u(x,t) be a solution of the Cauchy problem

     

Relaxation and regularity in the calculus of variations

In this work we prove that, if L(t,u,ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ, then any absolutely continuous solution to the variational problem

     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C¹$C^1$ solution u=u(t,x)$u=u(t,x)$ containing only n$n$ shock waves with small amplitude on t?0$t?0$ and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)$u=U(x/t)$ of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.     

New results of general n-dimensional incompressible Navier-Stokes equations

Let u=u(x,t,u₀) represent the global strong/weak solutions of the Cauchy problems for the general n-dimensional incompressible Navier-Stokes equations

     

Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary

This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. TMA 66 (1) (2007) 93-124]. In this paper, we study the global structure instability of the Riemann solution containing shocks, at least one rarefaction wave for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the nonexistence of global piecewise C¹ solution to a class of the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws on the quarter plane. Our result indicates that this kind of Riemann solution mentioned above for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable. Some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given.     

The utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term

The authors study the problem , and u(0,t)=u(1,t)=ψ(t), where ψ(t)=u₀ for t2k<t<t2k+1 and ψ(t)=0 for , with t₀=0 and the sequence tk is determined by the equations , for , and , for k=2,4,6,… and where 0<m<M. Note that the switching points , are unknown. Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1−tk are obtained and numerical verifications of the estimates are presented. The case of ux(0,t)=ux(1,t)=ψ(t) is also considered and analyzed.     

Hyperbolic systems of balance laws via vanishing viscosity

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension are constructed by the vanishing viscosity method of Bianchini and Bressan. For global existence, a suitable dissipativeness assumption has to be made on the production term g. Under this hypothesis, the viscous approximations u^?, that are globally defined solutions to , satisfy uniform BV bounds exponentially decaying in time. Furthermore, they are stable in L¹ with respect to the initial data. Finally, as ?→0, u^? converges in to the admissible weak solution u of the system of balance laws u_t+(f(u))_x+g(u)=0 when A=Df.     

Positive solutions and eigenvalue intervals of nonlocal boundary value problems with delays

The paper is concerned with the delay differential equation u^″+λb(t)f(u(t−τ))=0 satisfying u(t)=0 for −τ?t?0 and , where denotes the Riemann-Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g(t)=t, by using a property of the Riemann-Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established.     

Global existence for 2D nonlinear Schrödinger equations via high-low frequency decomposition method

We study global existence of solutions for the Cauchy problem of the nonlinear Schrödinger equation iut+Δu=|u|^2mu in the 2 dimension case, where m is a positive integer, m?2. Using the high-low frequency decomposition method, we prove that if then for any initial value φ∈Hs(R²), the Cauchy problem has a global solution in C(R,Hs(R²)), and it can be split into u(t)=eitΔφ+y(t), with y∈C(R,H¹(R²)) satisfying , where ? is an arbitrary sufficiently small positive number.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.