On the optimization of the initial boundary value problem for a conservation law

In the present note, the theory of shift differentiability for the Cauchy problem is extended to the case of an initial boundary value problem for a conservation law. This result allows to exhibit an Euler-Lagrange equation to be satisfied by the extrema of integral functionals defined on the solutions of initial boundary value problems of this kind.     

Well-posedness for entropy solutions to multidimensional scalar conservation laws with a strong boundary condition

In this paper we give a simple proof of well-posedness of multidimensional scalar conservations laws with a strong boundary condition. The proof is based on a result of strong trace for solutions of scalar conservation laws and kinetic formulation.     

The initial boundary value problem for quasi-linear wave equation with viscous damping

In this paper, the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping are proved. The nonexistence of the global solution for this problem is discussed by an ordinary differential inequality. Finally, an example is given.     

Existence of a positive solution of a fourth-order boundary value problem

In this paper, the existence of a positive solution of the boundary value problem of the following fourth-order nonlinear differential equation:

is discussed.     

Boundary layer to a system of viscous hyperbolic conservation laws

In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for n × n hyperbolic system of conservation laws with artificial viscosity in the half line （0, ∞）. We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.     

Solution of convex conservation laws in a strip

In this paper we consider scalar convex conservation laws in one space variable in a stripD =(x, t): 0 ≤x ≤1,t > 0 and obtain an explicit formula for the solution of the mixed initial boundary value problem, the boundary data being prescribed in the sense of Bardos-Leroux and Nedelec. We also get an explicit formula for the solution of weighted Burgers equation in a strip.     

一类二阶非线性奇异边值问题的正解

对非线性项ｆ在ｐｙ’＝ ０奇异但在ｙ＝ ０非奇异的情形下,边值问题 正解的存在性作讨论．     

Convergence of a continuous BGK model for conservation laws on the quarter plane

We consider a scalar conservation law in the quarter plan. This equation is approximated in a kinetic BGK model with infinite set of velocities. The convergence is established in the general BV framework, without special restrictions on the flux nor on the equilibrium problem's data.     

Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied：
x″（t）＋f（t,x（t））=0,0〈t〈1,
x′（0）=0,x（1）＋δx′（η）=0,
where η ∈ （0, 1）, δ∈ [0, ∞）, f ∈ C（[0, 1] × [0, ∞）, [0, ∞））. Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.     

Convergence rates of solutions toward boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space

This paper is concerned with the initial-boundary value problem of the generalized Benjamin-Bona-Mahony-Burgers equation in the half-space R₊

(I)     

Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws

This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws u_t + f(u)_x = u_xx on [0, 1], with the boundary condition u(0, t) = u₋(t) → u₋, u(1, t) = u₊(t) → u₊, as t → +∞ and the initial data u(x,0) = u₀(x) satisfying u₀(0) = u₋(0), u₀(1) = u₊(1), where u± are given constants, u₋ ≠ u₊ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When u₋ 〈 u₊, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u₋ > u₊, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, that is, |u₋−u₊| is small. Moreover, when u_±(t) ≡ u_±, t ≥ 0, exponential decay rates are both obtained.     

Upper and lower solution method for a fourth-order four-point boundary value problem on time scales

In this work, we consider a fourth-order four-point boundary value problem on time scales. We establish criteria for the existence of a solution by developing the upper and lower solution method and the monotone iterative technique.     

Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I

The shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations [A.M. Blokhin, On stability of shock waves in a compressible viscous gas, Matematiche LVII (I) (2002) 3-19]. We suggest such additional boundary conditions that a modified (with account to these conditions) linear initial-boundary value problem on stability of the shock wave does not admit Hadamard-type ill-posedness examples.     

Convergence of difference schemes with high resolution for conservation laws

We are concerned with the convergence of Lax-Wendroff type schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions-the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed estimates of the approximate solutions, compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws.

     

On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem

     

Nontrivial solution of a nonlinear second-order three-point boundary value problem

In this paper, for a second-order three-point boundary value problem u" f(t,u)=0, 0＜t＜l,au(0) - bu'(0) = 0, u(1) - αu(η) = 0,where η∈ (0, 1), a, b, α∈ R with a2 b2 ＞ 0, the existence of its nontrivial solution is studied.The conditions on f which guarantee the existence of nontrivial solution are formulated. As an application, some examples to demonstrate the results are given.