一类非散度型非线性退化抛物方程解的存在性(英文)

In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.     

Existence of solutions for a nonlocal epitaxial thin film growing equation

The existence of weak solutions is studied to the initial boundary problem of a nonlocal epitaxial thin film growing equation modeling epitaxial thin film growth. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.     

Some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary

In this paper, we study the very weak solutions to some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary. Firstly, we study the existence of the approximate solutions. Secondly, a priori estimates are given in the framework of weighted spaces. Finally, we prove the existence, uniqueness and regularity of the very weak solutions.     

具变指数非线性拟抛物方程弱解的存在性

虑一类变指数的非线性拟抛物方程的初值问题.在一些初值的假定下,基于时间离散化方法构造逼近解.通过对逼近解的一致性估计,证明了弱解的存在性.     

Existence and Uniqueness of Weak Solutions to the $p$-Biharmonic Parabolic Equation

We consider an initial-boundary value problem for a $p$-biharmonic parabolic equation. Under some assumptions on the initial value, we construct approximate solutions by the discrete-time method. By means of uniform estimates on solutions of the time-difference equations, we establish the existence of weak solutions, and also discuss the uniqueness.     

具变指数退化四阶抛物方程弱解的存在性

考虑一类具变指数退化四阶抛物方程的初边值问题.在一些初值的假定下,基于时间离散化方法构造逼近解,通过对逼近解的一致性估计,证明了弱解的存在性.     

Weak solutions in nonlinear poroelasticity with incompressible constituents

We consider quasi-static nonlinear poroelastic systems with applications in biomechanics and, in particular, tissue perfusion. The nonlinear permeability is taken to be dependent on solid dilation, and physical types of boundary conditions (Dirichlet, Neumann, and mixed) for the fluid pressure are considered. The system under consideration represents a nonlinear, implicit, degenerate evolution problem, which falls outside of the well-known implicit semigroup monotone theory. Previous literature related to proving existence of weak solutions for these systems is based on constructing solutions as limits of approximations, and energy estimates are obtained only for the constructed solutions. In comparison, in this treatment we provide for the first time a direct, fixed point strategy for proving the existence of weak solutions, which is made possible by a novel result on the uniqueness of weak solutions of the associated linear system (where the permeability is given as a function of space and time). The uniqueness proof for the associated linear problem is based on novel energy estimates for arbitrary weak solutions, rather than just for constructed solutions. The results of this work provide a foundation for addressing strong solutions, as well as uniqueness of weak solutions for nonlinear poroelastic systems.     

Existence and multiplicity of weak solutions for a singular semilinear elliptic equation

In this paper we study existence and multiplicity of weak solutions of the homogenous Dirichlet problem for a singular semilinear elliptic equation with a quadratic gradient term. The proofs for the main results are based on a priori estimates of solutions of approximate problems.     

一类拟线性椭圆边值变分问题及有限元分析

一、问题的提出 在工程科学中,例如在各种流动问题以及电磁场理论中,经常遇到下列非线性边值问题:     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

Global weak solutions to the Nordström-Vlasov system

The Nordström-Vlasov system is a Lorentz invariant model for a self-gravitating collisionless gas. We establish suitable a priori bounds on the solutions of this system, which together with energy estimates and the smoothing effect of “momentum averaging” yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion for classical solutions which was derived recently. The weak solutions are shown to preserve mass.     

Analysis of a parabolic cross-diffusion population model without self-diffusion

The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H¹ estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler-Galerkin approximation, discrete entropy estimates, and L¹ weak compactness arguments. Furthermore, employing the entropy-entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the cross-diffusion constants are.     

Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type

In this paper, we use the compensated compactness method with BV estimates on the Riemann invariants to obtain the global existence of bounded entropy weak solutions for the Cauchy problem of a symmetric system of Keyfitz-Kranzer type.     

Global existence and uniform decay for a nonlinear viscoelastic equation with damping

In this paper we investigate a nonlinear viscoelastic equation with linear damping. Global existence of weak solutions and the uniform decay estimates for the energy have been established.     

Global weak solution for a periodic generalized Hunter–Saxton equation

In this paper, we consider a periodic generalized Hunter–Saxton equation. We obtain the existence of global weak solutions to the equation. First, we give the well-posedness result of the viscous approximate equations and establish the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the equation.     

一类非线性渗流方程的Cauchy问题

讨论具有强非线性源和对流项的一般渗流方程以ＲＮ中某有界连续函数ｕ０（ｘ）或某一Ｒａｄｏｎ测度为初值的Ｃａｕｃｈｙ问题弱解的存在性，得到关于解的一系列重要估计．     

Existence,blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions

In this paper we consider a nonlinear Love equation associated with Dirichlet conditions. First, under suitable conditions, the existence of a unique local weak solution is proved. Next, a blow up result for solutions with negative initial energy is also established. Finally, a sufficient condition guaranteeing the global existence and exponential decay of weak solutions is given. The proofs are based on the linearization method, the Galerkin method associated with a priori estimates, weak convergence, compactness techniques and the construction of a suitable Lyapunov functional. To our knowledge, there has been no decay or blow up result for equations of Love waves or Love type waves before.     

On the analysis of an electronic model for solar cells including energy resolved defect densities

In this paper we study analytic properties of electronic models for solar cells, which take into account varying, energy resolved densities of defects. We establish energy estimates and the boundedness of weak solutions and give an existence and uniqueness result for weak solutions to the model equations in two space dimensions. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations

In this paper, we consider the Cauchy problem for the three dimensional chemotaxis-Navier–Stokes equations. By exploring the new a priori estimates, we prove the global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations.     

Well-posedness and dynamics for the fractional Ginzburg-Landau equation

This article studies the global well-posedness and long-time dynamics for the nonlinear complex Ginzburg–Landau equation involving fractional Laplacian. The global existence and some uniqueness criterion of weak solutions are given with compactness method. To study the strong solutions with the semigroup method, we generalize some pointwise estimates for the fractional Laplacian to the complex background and study carefully the linear evolution of the equation. Finally, the existence of global attractors is studied.