一类非线性绕射反应扩散系统

本文讨论了一类绕射反应扩散系统的初始边值问题.利用比较原理,研究了问题解的存在性,唯一性及其渐近性态.     

Global existence of strong solutions to the Cauchy problem for a 1D radiative gas

We consider a one-dimensional radiation hydrodynamics model in the case of the equilibrium diffusion approximation which is described by the compressible Navier-Stokes system with the additional terms in the pressure and internal energy respectively, which embody the effect of radiation. Under the physical growth conditions on the heat conductivity, we establish the existence and uniqueness of strong solutions to the Cauchy problem with large initial data, where the initial density and velocity may have differing constant states at infinity. Moreover, we show that if there is no vacuum in the initial density, then, the vacuum and concentration of the density will never occur in any finite time.     

Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Existence and uniqueness of positive solutions to a linear transport equation in a metric space

The linear functional equation ∂_tz = L(z) − rz is considered. The linear operator L acts on a linear metric space of real functions z depending on t and on a parameter ω belonging to a subset of ^m. The existence and uniqueness to a nonnegative solution of the initial value problem is shown. An application to a kinetic equation is performed.     

Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion

The existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory, a prominent example being the fundamental solutions of the linear theory. In nonlinear equations the existence of such solutions may find special obstacles, that can be either essential, or otherwise they may lead to more general concepts of solution. We give a particular review of results in the field of nonlinear diffusion.As a new contribution, we study in detail the case of logarithmic diffusion, associated with Ricci flow in the plane, where we can prove existence of measure-valued solutions. The surprising thing is that these solutions become classical after a finite time. In that general setting, the standard concept of weak solution is not adequate, but we can solve the initial-value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions. We prove that the problem is well-posed. The phenomenon of blow-down in finite time is precisely described: initial point masses diffuse into the medium and eventually disappear after a finite time Ti=Mi/4π.     

Global Existence of Large Data Weak Solutions for a Simplified Compressible Oldroyd-B Model Without Stress Diffusion

We start with the compressible Oldroyd-B model derived in [2] (J. W. Barrett, Y. Lu, and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15 (2017), 1265-1323), where the existence of global-in-time finite-energy weak solutions was shown in two dimensional setting with stress diffusion. In the paper, we investigate the case without stress diffusion. We first restrict ourselves to the corotational setting as in [28] (P. L. Lions, and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math., Ser. B, 21(2) (2000), 131-146). We further assume the extra stress tensor is a scalar matrix and we derive a simplified model which takes a similar form as the multi-component compressible Navier-Stokes equations, where, however, the pressure term related to the scalar extra stress tensor has the opposite sign. By employing the techniques developed in [30,35], we can still prove the global-in-time existence of finite energy weak solutions in two or three dimensions, without the presence of stress diffusion.     

Existence of multidimensional shock waves and phase boundaries

In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1-93] and Métivier's [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479] work. The linear stability results are based on Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95] work for the single shock front and Wang and Xin's [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529-558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479], we slightly modified Métivier's frame work to establish the existence for the solution to the nonlinear problem.     

Existence of multidimensional phase transitions in a non-isothermal van der Waals fluid

In this paper, we prove the existence of multidimensional subsonic phase transitions in a non-isothermal van der Waals fluid. The argument is based on the result of the existence of travelling waves given in [S.-Y. Zhang, Existence of travelling waves in non-isothermal phase dynamics, J. Hyperbolic Differ. Equ. 4 (3) (2007) 391–400] and the result of multidimensional stability given in [S.-Y. Zhang, Stability of multidimensional subsonic phase transition in a non-isothermal van der Waals fluid, Preprint]. An iteration technique [A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1–93] is applied to achieve the result.     

欧拉 - 玻尔兹曼方程整体光滑解

本文证明了R^d 中具有某一类小初值的等熵欧拉 - 玻尔兹曼方程整体光滑解的存在性.本文首先构造了等熵欧拉 - 玻尔兹曼方程的局部解, 并证明了局部解的适定性. 此外,文中还构造了关于原方程的随时间 t 增加、具有良好的衰减性质的整体光滑背景解. 同时, 当方程的辐射项系数满足一定条件时, 本文建立了关于源项的估计.通过将背景解的衰减与源项的估计结合起来, 文中证明了存在整数 s>d/2 + 1 ,使得背景解与原方程解的 H^s(R^d)x L²(R₊ x S^d-1;H^s(R^d))范数之差始终是有界的, 从而保证了原方程整体光滑解的存在性.     

The Global Existence of Small Solutions to the Oldroyd-B Model

The Cauchy problem to the Oldroyd-B model is studied. In particular, it is shown that if the smooth solution (u, τ) to this system blows up at a finite time T*, then ∫₀ ^T* ‖▿u(t)‖ _L ^∞dt = ∞. Furthermore, the global existence of smooth solution to this system is given with small initial data.     

平衡问题解的存在性

主要研究平衡问题解的存在性.通过对目标函数和可行集合的渐近分析,给出拟单调平衡问题解集非空的条件.进而用类似的方法研究了向量平衡问题解存在的条件,并将其应用到向量优化问题上.     

Global gradient estimates for degenerate parabolic equations in nonsmooth domains

This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.      

Banach空间中微分方程解的存在与唯一性

在一般Banach空间中,作者讨论了微分方程的初值问题u=f(t,u),u(0)=x0.在比文[6]中弱Carathodory条件更弱的情况下,不仅放宽了[6]中的一个重要不等式条件,还去掉了另一与之相关的不等式限制,仍获得了初值问题解的存在与唯一性及解的迭代逼近.对周期边值问题也获得了类似结果.     

Extragradient Methods for Vector Equilibrium Problems in Banach Spaces

We consider the vector equilibrium problem, which extends the scalar equilibrium problem to vector valued bifunctions, in a Banach space setting. We propose an extragradient method for solving this problem. Under suitable assumptions on the bifunction, we prove that the generated sequence is weakly convergent to a solution of the problem. Then, we propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the problem.     

具有功能反应的一类食物链交错扩散模型整体解的存在性

应用能量估计方法和Gagliardo-Nirenberg型不等式证明了具有Holling Ⅳ型功能反应的一类食物链模型在齐次Neumann边值条件下整体解的存在唯一性和一致有界性.     

一类耦合非线性波动方程全局解的存在性

利用微分不等式和解的延拓原理,研究了一类带非线性阻尼和源项的耦合非线性波动方程,通过分析方程中参数的关系,得到了全局解存在的一些新的充分条件.