Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

In this paper we discuss continuation properties and asymptotic behavior of -regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of -regular solutions is given. We also formulate sufficient conditions to construct a piecewise -regular solutions (continuation beyond maximal time of existence for -regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for in is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2mth order semilinear parabolic initial boundary value problem in a Hilbert space .     

Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities

This paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux. As the results of the interaction among the multi-coupled nonlinearities in the system, some appropriate conditions for global existence and global nonexistence of solutions are determined respectively.     

Local existence of solution to free boundary value problem for compressible Navier-Stokes equations

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local (in time) existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.     

Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density

In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(\rho_0,u_0)\in L^∞\times H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}\leq \varepsilon_0$ in bounded domain $\Omega \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(\rho_0,u_0)\in (W^{1,\gamma}∩L^∞)\times H^1_0$ in [13] and $(\rho_0,u_0)\in L^∞\times H^1_0$ in [7], which constitutes a positive answer to the question raised by Danchin and Mucha in [7]. The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach, and the continuity argument and shift of integrability method are applied to complete our proof.     

热方程的一些端点估计及其在Navier-Stokes方程中的应用——献给陆善镇教授75华诞

本文首先讨论热方程初值问题的解在Hardy、BMO（bounded mean oscillation）和Besov型空间中的估计.然后本文结合Coifmann-Lions-Meyer-Semmes在Hardy空间中的补偿紧性结果,给出Navier-Stokes方程整体弱解的二阶导数的一些端点估计.     

一维可压Navier—Stokes方程初边值问题局部解的存在性

我们证明了半空间中一维可压Navier—Stokes方程初边值问题局部解的存在性,证明主要是利用了能量方法．     

Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces . Though the case of the derivative index s=0 has been treated in our previous work, the case s≠0 is also treated in this paper. Our aim is to reveal the conditions of s, q and σ of for the existence of local and global solutions for initial data .     

The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

This paper deals with the critical exponents for the quasi-linear parabolic equations in Rn and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n?3, σ>−2/n and p>max{1,1+σ}, we obtain that pc=n(1+σ)/(n−2) is the critical exponent of these equations. Furthermore, we prove that if max{1,1+σ}<p?pc, then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f(x) and some initial data u₀(x) if p>pc. Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n=1,2, σ>−1 and p>max{1,1+σ}.     

Global solutions to kirchhoff equations with non-local nonlinearities defined by a potential

In this paper we investigate the global solvability for small initial data of Kirchhoff type equations is a suitable potential. Besides, we will give a partial characterization of the asymptotic behavior of the solutions as     

Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications

By a dual method, two Carleman estimates for forward and backward stochastic parabolic equations with Neumann boundary conditions are established. Then they are used to study a null controllability problem and a state observation problem for some stochastic forward parabolic equations with Neumann boundary conditions.     

Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities

This is a subsequent work of our previous one in [25]. Let the nonlinear terms of non-autonomous parabolic problems with singular initial data satisfy subcritical and critical growth conditions. We first establish the existence of uniform attractors in W^1,r(Ω)$W^{1,r}(\Omega )$, 1<r<N$1<r<N$, for the family of processes corresponding to the equations with external forces being translation bounded but not translation compact. Then, we prove the existence of pullback attractors in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively, for the process corresponding to the equation with the weaker assumption on the external force than previous one. Finally, we investigate the robust of attractors and establish the existence of pullback exponential attractors for the process acting in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively.     

Vacuum states on compressible Navier-Stokes equations with general density-dependent viscosity and general pressure law

In this paper,we study the one-dimensional motion of viscous gas with a general pres- sure law and a general density-dependent viscosity coefficient when the initial density connects to the vacuum state with a jump.We prove the global existence and the uniqueness of weak solutions to the compressible Navier-Stokes equations by using the line method.For this,some new a priori estimates are obtained to take care of the general viscosity coefficientμ(ρ)instead ofρ~θ.     

Global existence for some slightly super-linear parabolic equations with measure data

In this work we study the global existence of a solution to some parabolic problems whose model is

(1)     

Optimal control of nonlinear evolution equations with nonmonotone nonlinearities

In this paper, we prove the existence of optimal admissible pairs for a large class of strongly nonlinear evolution equations, involving nonmonotone nonlinearities. An example of a nonlinear parabolic optimal control system is also worked out in detail.The author wishes to thank Professor T. S. Angell for useful comments and suggestions.This research was supported by NSF Grant No. DMS-88-02688.     

Optimal control problems with state constraint governed by Navier-Stokes equations

This work deals with the existence of optimal solution and the maximum principle for optimal control problem governed by Navier-Stokes equations with state constraint of pointwise type in 3D. Strong results in 2D are also given.     

Initial-irregular oblique derivative problems for nonlinear parabolic systems of second order equations with measurable coefficients

The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and somea priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

Self-similar solutions to the isothermal compressible Navier-Stokes equations

^** Email: guo_zhenhua{at}iapcm.ac.cn^*** Email: jiang{at}iapcm.ac.cn We investigate the self-similar solutions to the isothermalcompressible Navier–Stokes equations. The aim of thispaper is to show that there exist neither forward nor backwardself-similar solutions with finite total energy. This generalizesthe results for the incompressible case in Neas, J., Rika, M.& verák, V. (1996, On Leray's self-similar solutionsof the Navier-Stokes equations. Acta. Math., 176, 283–294),and is consistent with the (unproved) existence of regular solutionsglobally in time for the compressible Navier–Stokes equations.     

Controllability for parabolic equations with uniformly bounded nonlinear terms

We consider a control system for a parabolic equation in a Banach space with uniformly bounded nonlinear termF,