SCALAR CONSERVATION LAWS ON A HALF-LINE: A PARABOLIC APPROACH

2010 ◽  
Vol 07 (01) ◽  
pp. 165-189 ◽  
Author(s):  
MIRIAM BANK ◽  
MATANIA BEN-ARTZI
Keyword(s):  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Scalar Conservation Laws ◽  
Duality Method ◽  
Scalar Conservation Law ◽  
Classical Duality ◽  
Scalar Conservation ◽  
Initial Boundary Value ◽  
Unique Limit

The initial-boundary value problem for the (viscous) nonlinear scalar conservation law is considered, [Formula: see text] The flux f(ξ) ∈ C2(ℝ) is assumed to be convex (but not strictly convex, i.e. f″(ξ)≥ 0). It is shown that a unique limit u = lim ∊ → 0 u∊ exists. The classical duality method is used to prove uniqueness. To this end parabolic estimates for both the direct and dual solutions are obtained. In particular, no use is made of the Kružkov entropy considerations.

ON LeFLOCH'S SOLUTIONS TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR CONSERVATION LAWS

2010 ◽  
Vol 07 (03) ◽  
pp. 503-543 ◽  
Author(s):  
HÉLÈNE FRANKOWSKA
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Conservation Laws ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Value Functions ◽  
Scalar Conservation Laws ◽  
Scalar Conservation ◽  
Initial Boundary Value

We consider the initial-boundary value problem for scalar conservation laws on the strip (0, ∞) × [0, 1] with strictly convex smooth flux of a superlinear growth. We show that an associated Hamilton–Jacobi equation with initial and (appropriately defined) boundary conditions has a unique generalized solution V that can be obtained as minimum of three value functions of the calculus of variation. Each of these functions, in turn, can be expressed using Lax's formula. The traces of the gradients Vx satisfy generalized boundary conditions (as in LeFloch (1988)) in a pointwise manner when the initial and boundary data are continuous and in a weak sense when they are discontinuous. It is also shown that Vx is continuous almost everywhere, and a result concerning the traces of the sign of f′(Vx(t, ⋅)) is proven.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

Sign in / Sign up

Export Citation Format

Share Document