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Explicit solution for the hyperbolic homogeneous scalar one-dimensional conservation law

Didier Clamond

Abstract

A complex integral formula provides an explicit solution of the initial value problem for the nonlinear scala 1D equation $u_t+[f(u)]_x = 0$, for any flux $f(u)$ and initial condition $u_0(x)$ that are analytic. This formula is valid at least as long as $u$ remains analytic.

Explicit solution for the hyperbolic homogeneous scalar one-dimensional conservation law

Abstract

A complex integral formula provides an explicit solution of the initial value problem for the nonlinear scala 1D equation , for any flux and initial condition that are analytic. This formula is valid at least as long as remains analytic.

Paper Structure

This paper contains 5 sections, 2 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Explicit solution for $c(u)$
  3. Explicit solution for $u(x,t)$
  4. Discussion
  5. Residue and Lagrange reversion theorems

Key Result

Theorem 1

If $u_0(x)$ and $c(u)=f'(u)$ are both real-analytic, and (at least) as long as $c(u(x,t))$ remains analytic, the initial value problem NLWE has the explicit solution for $c(u)$ where $\gamma$ is a rectifiable Jordan curve surrounding $x$ such that $c\circ u_0$ is holomorphic inside $\gamma$ and $|z-x|>|t\,c(u_0(z))|$ on $\gamma$.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof