Well-posedness of a model of point dynamics for a limit of the Keller-Segel system

The purpose of this paper is to prove well-posedness for a problem that describes the dynamics of a set of points by means of a system of parabolic equations. It has been seen in Velázquez (Point dynamics in a singular limit of the Keller-Segel model. (1) motion of the concentration regions, SIAM J. Appl. Math., to appear) that the considered model is the limit of a singular perturbation problem for a system of the Keller-Segel type.     

Well-posedness results and asymptotic behaviour for a phase transition model taking into account microscopic accelerations

We obtain an existence and uniqueness result to Frémond's phase transition model which take into account microscopic movements and accelerations. Moreover, the irreversible evolution of the phase variable is considered. Next, we perform an asymptotic analysis on the solution to the above problem, as the power of the microscopic acceleration forces goes to zero.     

Well-posedness and stability results for a quasilinear periodic Muskat problem

We study the Muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two-dimensional geometry. We first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown. Secondly, we show that this evolution equation has a quasilinear structure, which is at a formal level not obvious, and we also disclose the parabolic character of the equation. Exploiting these aspects, we establish the local well-posedness of the problem for arbitrary initial data in $H^s(\mathbb{S})$, with $s\in (3/2,2)$, determine a new criterion for the global existence of solutions, and uncover a parabolic smoothing property. Besides, we prove that the zero steady-state solution is exponentially stable.     

Finite element approximation to a contact problem for a nonlinear thermoviscoelastic beam

In this paper we propose and analyze a finite element method to the solution of a quasi-static contact problem between a nonlinear beam and a rigid obstacle. Error estimates and energy decay are obtained and some numerical simulations described.     

Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow

In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class $2?\theta$ for any $\theta >0$. This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like $e^{\sqrt{k}t}$ for the linearized Prandtl equation around a non-monotonic shear flow.     

Well-posedness and the small time large deviations of the stochastic integrable equation governing short-waves in a long-wave model

This paper is concerned with the stochastic integrable equation governing short-waves in a long-wave model. Firstly, the local well-posedness for this system is established by fixed point argument and (bilinear) trilinear estimates. Then the small time asymptotics of the equation is proved. The corresponding results for the stochastic Hunter–Saxton equation can be obtained by the same methods.     

Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint

In this paper, we study the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems and scalar equilibrium problems as constraint.     

Analysis of a system related to a model for phase transitions in dissipative isochoric thermoviscoelastic materials

We analyze a highly nonlinear system of partial differential equations related to a model solidification and/or melting of thermoviscoelastic isochoric materials with the possibility of motion of the material during the process. This system consists of an internal energy balance equation governing the evolution of temperature, coupled with an evolution equation for a phase field whose values describe the state of material and a balance equation for the linear moments governing the material displacements. For this system, under suitable dissipation conditions, we prove global existence and uniqueness of weak solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

耦合Schrdinger-KdV方程组Cauchy问题的适定性

本文研究了耦合Schrodinger－KdV方程组的Cauchy问题，此耦合方程组刻化了一维Langmuir和离子声波相互作用的非线性动力学行为．本文建立了此问题在Hk×Hk中的整体适定性理论（k∈Z+）．     

Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors

In this paper we study a free boundary problem modelling the growth of nonnecrotic tumors. The main trait of this free boundary problem is that it is essentially multidimensional, so that its well-posedness is hard to establish by using the usual methods in the classical theory of free boundary problems. In this paper we use the functional analysis method based on the theory of analytic semigroups to prove that this problem has a unique local solution in suitable function spaces. Continuous dependence of the solution on the initial data and regularities of the solution can also be easily obtained by using the argument of this paper.     

2维耗散准地转方程在齐次Morrey型Besov空间的适定性(英文)

本文讨论2维耗散准地转方程在齐次Morrey型Besov空间的初值问题.首先建立齐次Morrey型Besov空间的一个新特征,然后利用此特征和Kato方法,证明当初始值以齐次Morrey型Besov空间内的范数很小时,2维耗散准地转方程对时间的全局解的存在性和唯一性.     

Existence for a thermoviscoelastic beam model of brakes

The existence of a weak solution to a model for the dynamic thermomechanical behavior of a viscoelastic beam, which is in frictional contact with a rigid rotating wheel, is established. The model describes a simple braking system in which a rotating wheel comes to a stop as a result of the frictional traction generated by the beam. The classical model consists of a system of coupled equations for the beam temperature and displacement, the wear of the beam's contacting end, the wheel temperature and its angular velocity. The weak formulation is an abstract differential inclusion involving set-valued pseudomonotone operators, The existence is proved by using recent results for such operators. Uniqueness is shown to hold when the wheel's angular velocity and temperature are known.     

Well-posedness of the free boundary problem in compressible elastodynamics

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.     

Global Well-posedness for an incompressible flow with intrinsic degrees of freedom in bounded domains

We consider a mathematical model for an incompressible Newtonian fluid with intrinsic degrees of freedom in a smooth bounded domain. We first show that there exists a unique local strong solution for large initial data. Then, we prove that the local strong solution is indeed global provided that the initial data is sufficiently small. Furthermore, we prove that when the strong solution exists, all the global weak solutions constructed by Lions must be equal to the unique strong solution with the same initial data.     

故障时间任意分布的两部件可修复系统解的适定性分析

讨论了一个由两个部件并联组成的可修复冗余系统模型,修复后的故障系统恢复如新.在假设修复函数有界的条件下,给出了C_0-半群的生成元(系统算子)对应的柯西问题的解的适定性分析.     

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.     

On the Cauchy problem for a model equation for shallow water waves of moderate amplitude

We prove the local well-posedness for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime.     

Well-posedness of the Cauchy problem of Ostrovsky equation in anisotropic Sobolev spaces

We study the Cauchy problem of the Ostrovsky equation , with βγ<0. By establishing a bilinear estimate on the anisotropic Bourgain space Xs,ω,b, we prove that the Cauchy problem of this equation is locally well-posed in the anisotropic Sobolev space H(s,ω)(R) for any and some . Using this result and conservation laws of this equation, we also prove that the Cauchy problem of this equation is globally well-posed in H(s,ω)(R) for s?0.     

Well-posedness by perturbations of mixed variational inequalities in Banach spaces

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a mixed variational inequality problem in a Banach space. We establish some metric characterizations of the well-posedness by perturbations. We also show that under suitable conditions, the well-posedness by perturbations of a mixed variational inequality problem is equivalent to the well-posedness by perturbations of a corresponding inclusion problem and a corresponding fixed point problem. Also, we derive some conditions under which the well-posedness by perturbations of a mixed variational inequality is equivalent to the existence and uniqueness of its solution.