A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

In this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.     

Uniqueness Result in the Cauchy Dirichlet Problem via Mehler Kernel

Using the Mehler kernel, a uniqueness theorem in the Cauchy Dirichlet problem for the Hermite heat equation with homogeneous Dirichlet boundary conditions on a class P of bounded functions U(x, t) with certain growth on U _x (x, t) is established.     

Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ ^N . The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In the case of the symmetric group S _N , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments. Received: 10 March 1997 / Accepted: 7 July 1997     

On the Cauchy Problem for a Nonlinearly Dispersive Wave Equation

Abstract

We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite time. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.     

On the Cauchy Problem for the Inelastic Boltzmann Equation with External Force

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered for near vacuum data. Under the assumptions on the bicharacteristic generated by external force which can be arbitrarily large, we prove the global existence of mild solution for initial data small enough with respect to the sup norm with exponential weight by using the contraction mapping theorem. Furthermore, we prove the uniform L ¹ stability of the mild solution following from the exponential decay estimate and the Gronwall’s inequality for the case of soft potentials.     

Cauchy Problem for the Fermion Field Equation Coupled With the Chern–Simons Gauge

We study initial value problems of the Chern–Simons–Dirac equations. With the Lorentz gauge condition they are formulated in the second-order hyperbolic equations. Under the Coulomb gauge condition Dirac equation is coupled with the elliptic equations which show some smoothing properties of the gauge field. With the temporal gauge condition divergence-curl decomposition and elliptic estimates will be used. JSPS Research Follow supported by JSPS Grant-in-Aid     

A Two-Phase Free Boundary Problem for the Nonlinear Heat Equation

Abstract

A two-phase free boundary problem associated with nonlinear heat conduction is considered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution is proved for small times and a particular exact solution is discussed.     

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.     

A Method for Studying the Cauchy Problem for a Singularly Perturbed Weakly Nonlinear First-Order Differential Equation

A sequence converging to the solution of the Cauchy problem for a singularly perturbed weakly nonlinear first-order differential equation is constructed. This sequence is asymptotic in the sense that the distance (with respect to the norm of the space of continuous functions) between its nth element and the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter. Such a sequence can be used to justify asymptotics obtained by the boundary function method.     

Global Solutions to the Cauchy Problem for the Relativistic Boltzmann Equation with Near–Vacuum Data

The Cauchy Problem for the relativistic Boltzmann equation is studied with small (i.e., near–vacuum) data. For an appropriate class of scattering cross sections, global ``mild' solutions are obtained. Supported in part by NSF DMS 0204227     

Analysis of the Generalized Heat Equation for Solution of the Dynamic Thermoelasticity Problem

Doklady Physics - On the basis of the approximate solution of the dispersion equation, the set of equations of dynamic thermoelasticity is analyzed with taking into account the generalized heat...     

Existence and Uniqueness of Solutions for the Couette Problem

We study existence and uniqueness results for the one-dimensional Boltzmann equation with inflow and diffusive boundary conditions. Our focus, partly encompasses some of the properties of the Boltzmann collision gain term which play a significant role in existence and uniqueness results. A series of estimates are proven on the collision term which is shown to produce a suitable function space in which the contraction mapping arguments are available.This work was done as part of the authors M.Sc. studies at the University of Victoria.     

On Uniqueness in the General Inverse Transmisson Problem

In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.     

Existence and Uniqueness of Low Regularity Solutions for the Dullin-Gottwald-Holm Equation

We establish the local existence and uniqueness of solutions for the Dullin-Gottwald-Holm equation with continuously differentiable, periodic initial data. The regularity conditions needed for the Cauchy problem via the semigroup approach of quasilinear hyperbolic equations of evolution or the viscosity method are significantly lowered.     

On Stable Wall Boundary Conditions for the Hermite Discretization of the Linearised Boltzmann Equation

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331–407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.     

On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation

In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as ²0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.     

Uniqueness Problem for Quantum Lattice Systems with Compact Spins

Quantum lattice systems with compact spins and nearest-neighbour interactions are considered. Uniqueness of the corresponding Euclidean Gibbs states is proved uniformly with respect to the temperature, in the case where the particles have a sufficiently small mass.     

Uniqueness of Bounded Solutions for the Homogeneous Landau Equation with a Coulomb Potential

We prove the uniqueness of bounded solutions for the spatially homogeneous Fokker-Planck-Landau equation with a Coulomb potential. Since the local (in time) existence of such solutions has been proved by Arsen’ev–Peskov (Z. Vycisl. Mat. i Mat. Fiz. 17:1063–1068, 1977), we deduce a local well-posedness result. The stability with respect to the initial condition is also checked.