GLOBAL SOLUTIONS TO THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE HEAT-CONDUCTING FLOW WITH SYMMETRY AND FREE BOUNDARY

ABSTRACT

Global solutions of the multidimensional Navier-Stokes equations for compressible heat-conducting flow are constructed, with spherically symmetric initial data of large oscillation between a static solid core and a free boundary connected to a surrounding vacuum state. The free boundary connects the compressible heat-conducting fluids to the vacuum state with free normal stress and zero normal heat flux. The fluids are initially assumed to fill with a finite volume and zero density at the free boundary, and with bounded positive density and temperature between the solid core and the initial position of the free boundary. One of the main features of this problem is the singularity of solutions near the free boundary. Our approach is to combine an effective difference scheme to construct approximate solutions with the energy methods and the pointwise estimate techniques to deal with the singularity of solutions near the free boundary and to obtain the bounded estimates of the solutions and the free boundary as time evolves. The convergence of the difference scheme is established. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.     

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT

We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.     

ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT

We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.     

Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

A free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated. The existence, uniqueness, and regularity of global solutions are established with large initial data in H¹. It is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. An existence theorem of global solutions with large discontinuous initial data is also established.     

Singular Perturbation in a Nonlocal Diffusion Problem

We study a singular perturbation problem for a nonlocal evolution operator. The problem appears in the analysis of the propagation of flames in the high activation energy limit, when admitting nonlocal effects.

We obtain uniform estimates and we show that, under suitable assumptions, limits are solutions to a free boundary problem in a viscosity sense and in a pointwise sense at regular free boundary points.

We study the nonlocal problem both for a single equation and for a system of two equations.

Some of the results obtained are new even when the operator under consideration is the heat operator.     

ALMOST OPTIMAL LOCAL WELL-POSEDNESS OF THE MAXWELL-KLEIN-GORDON EQUATIONS IN 1 + 4 DIMENSIONS

Abstract

We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ? R ³. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R ³) or the initial boundary value problem (for Ω ? ? R ³) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.     

Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper,the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied.The solutions to the system(1.6)–(1.8) are in the class of radius-dependent solutions,i.e.,independent of the axial variable and the angular variable.In particular,the expanding rate of the moving boundary is obtained.The main difficulty of this problem lies in the strong coupling of the magnetic field,velocity,temperature and the degenerate density near the free boundary.We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates,and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable r;by weighted estimates,and also the uniform-in-time weighted estimates of the higher-order derivatives of solutions by delicate analysis.     

An Initial Boundary-Value Problem in a Half-Strip for the Korteweg–De Vries Equation in Fractional-Order Sobolev Spaces

Abstract

An initial boundary-value problem in a half-strip with one boundary condition for the Korteweg–de Vries equation is considered and results on global well-posedness of this problem are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a “boundary potential” type for this linearized KdV equation.     

Parabolic Ito Equations with Mixed in Time Conditions

Abstract

We study linear stochastic partial differential equations of parabolic type with special boundary conditions in time. The standard Cauchy condition at the initial time is replaced by a condition that mixes the values of the solution at different times, including the terminal time and continuously distributed times. Uniqueness, solvability and regularity results for the solutions are obtained.     

On quasilinear parabolic evolution equations in weighted L p -spaces

In this paper we develop a geometric theory for quasilinear parabolic problems in weighted L _p -spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) L _p -spaces.     

Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data

We consider the nonlinear heat equation (with Leray-Lions operators) on an open bounded subset of R^N with Dirichlet homogeneous boundary conditions. The initial condition is in L¹ and the right hand side is a smooth measure. We extend a previous notion of entropy solutions and prove that they coincide with the renormalized solutions.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Unloading near a shear band: a free boundary problem for the wave equation

ABSTRACT

For solutions on unbounded domains of boundary value problems for a class of quasilinear elliptic equations which are not uniformly elliptic, we prove that the solutions have the same bounds as those of the boundary data.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

A Free Boundary Problem of Fourth Order: Classical Solutions in Weighted Hölder Spaces

We prove short-time existence and uniqueness of classical solutions in weighted Hölder spaces for the thin-film equation with linear mobility, zero contact angle, and compactly supported initial data. We furthermore show regularity of the free boundary and optimal regularity of the solution in terms of the regularity of the initial data. Our approach relies on Schauder estimates for the operator linearized at the free boundary, obtained through a variant of Safonov's method that is solely based on energy estimates.     

Evolution of nonparametric surfaces with speed depending on curvature II. The mean curvature case

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞.     

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u_tt?u_xx=b(u,Du)u_xx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem 1.1 is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span 1.8 in the general case and 1.10 in the special case. Copyright © 2015 John Wiley & Sons, Ltd.