In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma...In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-′qF(q) + L(′F(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T~*. Meanwhile, we can give the equation of the shock and an explicit value of T~*.展开更多
We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension con...We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.展开更多
Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new...Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.展开更多
The paper studies an evolutionary p(x)-Laplacian equation with a convection term ut=div(ρα|■u|p(x)-2■u)+∑N i=1■bi(u)/■xi,whereρ(x)=dist(x,■Ω),ess inf p(x)=p^->2.To assure the well-posedness of the solutio...The paper studies an evolutionary p(x)-Laplacian equation with a convection term ut=div(ρα|■u|p(x)-2■u)+∑N i=1■bi(u)/■xi,whereρ(x)=dist(x,■Ω),ess inf p(x)=p^->2.To assure the well-posedness of the solutions,the paper shows only a part of the boundary,Σp■■Ω,on which we can impose the boundary value.Σp is determined by the convection term,in particular,when 1<α<(p^--2)/2,Σp={x∈■Ω:bi′(0)ni(x)<0}.So,there is an essential difference between the equation and the usual evolutionary p-Laplacian equation.At last,the existence and the stability of weak solutions are proved under the additional conditionsα<(p^--2)/2 andΣp=■Ω.展开更多
基金supported in part by NSFC(11671193)the Fundamental Research Funds for the Central Universities(NE2015005)supported in part by NSFC(11271182 and 11501290)
文摘In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-′qF(q) + L(′F(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T~*. Meanwhile, we can give the equation of the shock and an explicit value of T~*.
基金partially supported by the Natural Science Foundation of China(11271105)a grant from the China Scholarship Council and a Humboldt fellowship of Germany
文摘We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.
基金supported by the National Natural Science Foundation of China(No.11371297)the Science Foundation of Xiamen University of Technology(No.XYK201448)
文摘Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.
基金Supported by the National Natural Science Foundation of China(No.2015J01592,No.2019J01858)
文摘The paper studies an evolutionary p(x)-Laplacian equation with a convection term ut=div(ρα|■u|p(x)-2■u)+∑N i=1■bi(u)/■xi,whereρ(x)=dist(x,■Ω),ess inf p(x)=p^->2.To assure the well-posedness of the solutions,the paper shows only a part of the boundary,Σp■■Ω,on which we can impose the boundary value.Σp is determined by the convection term,in particular,when 1<α<(p^--2)/2,Σp={x∈■Ω:bi′(0)ni(x)<0}.So,there is an essential difference between the equation and the usual evolutionary p-Laplacian equation.At last,the existence and the stability of weak solutions are proved under the additional conditionsα<(p^--2)/2 andΣp=■Ω.
