REGULARITY OF SOLUTIONS TO CHARACTERISTIC INITIAL-BOUNDARY VALUE PROBLEMS FOR SYMMETRIZABLE SYSTEMS

2009 ◽  
Vol 06 (04) ◽  
pp. 753-808 ◽  
Author(s):  
ALESSANDRO MORANDO ◽  
PAOLO SECCHI ◽  
PAOLA TREBESCHI
Keyword(s):  
Energy Estimate ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regularity Of Solutions ◽  
Constant Rank ◽  
Initial Boundary Value Problems ◽  
Characteristic Boundary ◽  
Initial Boundary Value ◽  
Structural Assumption

We consider the initial-boundary value problem for linear Friedrichs symmetrizable systems with characteristic boundary of constant rank. We assume the existence of the strong L2 solution satisfying a suitable energy estimate, but we do not assume any structural assumption sufficient for existence, such as the fact that the boundary conditions are maximally dissipative or the Kreiss–Lopatinski condition. We show that this is enough in order to get the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth.

A class of densely invertible parabolic operator equations

1969 ◽  
Vol 1 (3) ◽  
pp. 363-374 ◽  
Author(s):  
R.S. Anderssen
Keyword(s):  
Variational Methods ◽  
Variational Formulation ◽  
Lagrange Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Parabolic Differential Equations ◽  
Operator Equations ◽  
Initial Boundary Value

Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.

Aspects of Osillations of Beams and Plates

2005 ◽  
Author(s):  
Mariya A. Zarubinska ◽  
W. T. van Horssen
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Boundary Values ◽  
Internal Resonances ◽  
Specific Parameter ◽  
Plate Equations ◽  
Initial Boundary Value ◽  
Parameter Values

In this paper some initial boundary value problems for beam and plate equations will be studied. These initial boundary values problems can be regarded as simple models describing free oscillations of plates on elastic foundations or describing coupled torsional and vertical oscillations of a beam. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For the plate on an elastic foundation it turns out that complicated internal resonances can occur for specific parameter values.

scholarly journals Exact variational representations for the solution of the simple parabolic equation

1971 ◽  
Vol 5 (3) ◽  
pp. 305-314
Author(s):  
R.S. Anderssen
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Differential Equation ◽  
Initial Boundary Value Problems ◽  
Series Representations ◽  
Initial Boundary Value

By constructing a special set of A-orthonormal functions, it is shown that, under certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.

THE DISCRETE BOLTZMANN EQUATION: A REVIEW OF THE MATHEMATICAL ASPECTS OF THE INITIAL AND INITIAL-BOUNDARY VALUE PROBLEMS

1991 ◽  
Vol 03 (02) ◽  
pp. 137-162 ◽  
Author(s):  
N. BELLOMO ◽  
T. GUSTAFSSON
Keyword(s):  
Boltzmann Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Future Research ◽  
Initial Boundary Value Problems ◽  
Survey Paper ◽  
Initial Boundary Value ◽  
Mathematical Literature ◽  
Discrete Boltzmann Equation

This paper provides a review of the various results available in the mathematical literature on the solution to the initial and initial-boundary value problem for the discrete Boltzmann equation, a nonlinear mathematical model of the kinetic theory of gases. References include papers published until 1990. The aim of this survey paper is to provide a detailed picture of the state of the art in the field and some necessary information for future research in the field.

Initial-boundary value problems for linear PDEs with variable coefficients

2007 ◽  
Vol 143 (1) ◽  
pp. 221-242 ◽  
Author(s):  
P. A. TREHARNE ◽  
A. S. FOKAS
Keyword(s):  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Initial Boundary Value Problems ◽  
Linear Evolution ◽  
Higher Derivatives ◽  
Initial Boundary Value

AbstractA new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

Bounds for the solution of a non-linear parabolic initial-boundary value problem

1977 ◽  
Vol 82 (1) ◽  
pp. 131-145
Author(s):  
M. R. Carter
Keyword(s):  
Steady State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
The Past ◽  
Existence And Stability ◽  
Image Position ◽  
Initial Boundary Value ◽  
Steady State Solutions

A number of papers have appeared over the past decade or so which study questions of the existence and stability of positive steady-state solutions for parabolic initial-boundary value problems of the general form

Abstract initial boundary value problems

1994 ◽  
Vol 124 (5) ◽  
pp. 879-908 ◽  
Author(s):  
C. Palencia ◽  
I. Alonso Mallo
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Operators ◽  
Initial Boundary Value Problems ◽  
Hyperbolic Problems ◽  
Euler’S Method ◽  
Initial Boundary Value

We consider abstract initial boundary value problems in a spirit similar to that of the classical theory of linear semigroups. We assume that the solution u at time t is given by u(t) = S(t) ξ + V(t)g, where ξ and g are respectively the initial and boundary data and S(t) and V(t) are linear operators. We take as a departing point the functional equations satisfied by the propagators S and V. We discuss conditions under which a pair (S, V) describes the solution of an abstract differential initial boundary value problem. Several examples are provided of parabolic and hyperbolic problems that can be accommodated within the abstract theory. We study the backward Euler's method for the time integration of the problems considered.

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