Well-posedness of the free boundary problem for incompressible elastodynamics

The free boundary problem for the three dimensional incompressible elastodynamics system is studied under the Rayleigh–Taylor sign condition. Both the columns of the elastic stress $\mathbf{F}{\mathbf{F}}^{?}?\mathrm{I}$ and the transpose of the deformation gradient ${\mathbf{F}}^{?}?\mathrm{I}$ are tangential to the boundary which moves with the velocity, and the pressure vanishes outside the flow domain. The linearized equation takes the form of wave equation in terms of the flow map in the Lagrangian coordinate, and the local-in-time existence of a unique smooth solution is proved using a geometric argument in the spirit of [19].     

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

Global existence and blowup of a localized problem with free boundary

This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large.     

Supremum principles for the higher-order derivatives of solutions to the Dirichlet problem for parabolic equations without compatibility conditions between the data

The Dirichlet problem is considered for the heat equation ut=auxx, a>0 a constant, for (x,t)∈[0,1]×[0,T], without assuming any compatibility condition between initial and boundary data at the corner points (0,0) and (1,0). Under some smoothness restrictions on the data (stricter than those required by the classical maximum principle), weak and strong supremum and infimum principles are established for the higher-order derivatives, ut and uxx, of the bounded classical solutions. When compatibility conditions of zero order are satisfied (i.e., initial and boundary data coincide at the corner points), these principles allow to estimate the higher-order derivatives of classical solutions uniformly from below and above on the entire domain, except that at the two corner points. When compatibility conditions of the second order are satisfied (i.e., classical solutions belong to on the closed domain), the results of the paper are a direct consequence of the classical maximum and minimum principles applied to the higher-order derivatives. The classical principles for the solutions to the Dirichlet problem with compatibility conditions are generalized to the case of the same problem without any compatibility condition. The Dirichlet problem without compatibility conditions is then considered for general linear one-dimensional parabolic equations. The previous results as well as some new properties of the corresponding Green functions derived here allow to establish uniformL¹-estimates for the higher-order derivatives of the bounded classical solutions to the general problem.     

On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD

We study the initial-boundary value problem resulting from the linearization of the plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). We suppose that the plasma and the vacuum regions are unbounded domains and the plasma density does not go to zero continuously, but jumps. For the basic state upon which we perform linearization we find two cases of well-posedness of the “frozen” coefficient problem: the “gas dynamical” case and the “purely MHD” case. In the “gas dynamical” case we assume that the jump of the normal derivative of the total pressure is always negative. In the “purely MHD” case this condition can be violated but the plasma and the vacuum magnetic fields are assumed to be non-zero and non-parallel to each other everywhere on the interface. For this case we prove a basic a priori estimate in the anisotropic weighted Sobolev space for the variable coefficient problem.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.     

Existence and uniqueness for a nonlinear discrete hyperbolic system

The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system subject to some extreme conditions and initial data are investigated in a real Hilbert space.     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C¹ solutions are obtained for small initial and boundary data. We also present two applications for physical models.     

On a boundary value problem in the half-space

In this paper, we study the existence of positive solutions and sign-changing solutions for the following boundary value problem in the half-space

     

半线性波动方程的高维古沙问题

In this paper we study the Goursat problem for semilinear wave equations with zero boundary condition in which the boundary is the characteristic cone for wave operator. Our result states that the solution is Lipschitz and is smooth awayfrom the characteristic cone.     

Reconstruction of two time independent coefficients in an inverse problem for a phase field system

In this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain Ω⊂Rⁿ for the dimension n≤3 with a single observation on a subdomain ω?Ω and the Sobolev norm of certain partial derivatives of the solutions at a fixed positive time θ∈(0,T) over the whole spatial domain. The proof of these results relies on an appropriate Carleman estimate for the phase field system.     

NOTE ON A BOUNDARY VALUE PROBLEM OF POISSON＇S EQUATION

This note shows that fur a BVP of Poisson‘s equation with Qm(u,v) as its source function, a direct evaluation of some integrals yields the same exact results as obtained by similarity analysis.     

Cauchy–Dirichlet problem for second-order hyperbolic equations in cylinders with non-smooth base

This paper is concerned with the smoothness of generalized solutions of the Cauchy–Dirichlet problem for the second-order hyperbolic equation in domains with a conical point.     

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.     

On the solutions of the Cauchy problem for a class of partial differential equations with double characteristic at a point

We give an explicit representation of the solutions of the Cauchy problem, in terms of series of hypergeometric functions, for the following class of partial differential equations with double characteristic at the origin:

(xkt_∂+ax_∂)(xkt_∂+bx_∂)u+cxk−1t_∂u=0,     

Initial value problems for a system of nonlinear equations on a half-line

The paper deals with initial value problems for the system of nonlinear equations on a half-line where ₁, ₁ are real constant connections.The first problem for the nonlinear Schrödinger equation (NLS) is obtained from the system (A) when . In this case, the boundary problem associated with this NLS does not have a discrete spectrum, and the solution of the NLS is uniquely found from the given initial condition. Further, we show two remarkable classes of matrix potentials, in which the inverse scattering problem associated with system (A) can be solved exactly. In the case where the given scattering data for (A) consist of only two simple poles, the exact soliton solution of system (A) is presented. This happens if and only if the time evolution of the scattering data for (A) obeys some time evolution equations. In this case, the initial value problem for the NLS which is obtained from (A) when is solved exactly.     

Oscillation criteria for boundary value problems of high-order partial functional differential equations

In this paper, a class of boundary value problems associated with high-order partial functional differential equations with distributed deviating arguments is investigated. Some oscillation criteria of solutions to the problem are developed. Our approach is to reduce the multi-dimensional oscillation problem to a one-dimensional oscillation one by employing some integral means of solutions and introducing some parameter functions. One illustrative example is considered.     

Well-posedness and blow-up phenomena for a higher order shallow water equation

This paper deals with the Cauchy problem for a higher order shallow water equation yt+auxy+buyx=0, where and k=2. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R) with s?7/2 is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered.     

A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation

We present a systematic study of local solutions of the ODE of the form near t=0. Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ>0 and γ<0, where γ=∂f/∂x^′ at t=0. As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.