Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

Validity of Nonlinear Geometric Optics for Entropy Solutions of Multidimensional Scalar Conservation Laws

Nonlinear geometric optics with various frequencies for entropy solutions only in L∞ of multidimensional scalar conservation laws is analyzed. A new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, and L–stability. New multidimensional features are recognized, especially including nonlinear propagations of oscilla...

Structure of Entropy Solutions for Multi{dimensional Scalar Conservation Laws

An entropy solution u of a multi{dimensional scalar conservation law is not necessarily in BV , even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV {function in the sense that the shock location is codimension{one rectiiable. This result highlights the regularizing eeect of genuine non-linearity in a qualitative way; it is based on the loca...

Error Estimates of Approximate Solutions for Nonlinear Scalar Conservation Laws

There has been an enormous amount of work on error estimates for approximate solutions to scalar conservation laws. The methods of analysis include matching the traveling wave solutions, [8, 24]; matching the Green function of the linearized problem [21]; weak W convergence theory [32]; the Kruzkov-functional method [19]; and the energy-like method [34]. The results on error estimates include: ...

On weak convergence of entropy solutions to scalar conservation laws

We prove that weak limits of entropy solutions to a one-dimensional scalar conservation law are entropy solutions as well. We consider a scalar conservation law ut + f(u)x = 0, (t, x) ∈ Π = (0, +∞)× R. (1) The flux function f(u) is supposed to be only continuous: f(u) ∈ C(R). Recall the notion of an entropy solution of (1) in the sense of Kruzhkov [6]. Definition 1. A bounded measurable functio...

L2 formulation of multidimensional scalar conservation laws