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Viscous Burgers equation driven by point source: a formula for the weak limit

Smritikana Pal, Manas R. Sahoo

TL;DR

This work analyzes the viscous Burgers equation driven by a point source and derives the weak vanishing-viscosity limit as $\epsilon\to0$. Via the Hopf–Cole transform, explicit viscous solutions on each side of the origin are obtained and linked by a boundary term, yielding tractable representations and a corresponding weak formulation. The vanishing viscosity limit is characterized by the minimum of three variational functionals on each half-line, producing explicit limit functionals $U_R$ and $U_L$ whose derivatives give the limiting velocity field. The limit $u(x,t)$ satisfies the inviscid equation $u_t+(u^2/2)_x=\delta$ in the distributional sense, establishing a rigorous connection between viscous regularization and a weak solution for a scalar balance law with a delta source and discontinuous flux.

Abstract

In this article, we obtain the weak limit of the solutions of the viscous Burgers equation driven by a point source term, as the coefficient of viscosity tends to zero. The weak limit is related to the variational problem that consists of three types of functional, which is not usual in the absence of the source term.

Viscous Burgers equation driven by point source: a formula for the weak limit

TL;DR

This work analyzes the viscous Burgers equation driven by a point source and derives the weak vanishing-viscosity limit as . Via the Hopf–Cole transform, explicit viscous solutions on each side of the origin are obtained and linked by a boundary term, yielding tractable representations and a corresponding weak formulation. The vanishing viscosity limit is characterized by the minimum of three variational functionals on each half-line, producing explicit limit functionals and whose derivatives give the limiting velocity field. The limit satisfies the inviscid equation in the distributional sense, establishing a rigorous connection between viscous regularization and a weak solution for a scalar balance law with a delta source and discontinuous flux.

Abstract

In this article, we obtain the weak limit of the solutions of the viscous Burgers equation driven by a point source term, as the coefficient of viscosity tends to zero. The weak limit is related to the variational problem that consists of three types of functional, which is not usual in the absence of the source term.
Paper Structure (4 sections, 11 theorems, 133 equations)

This paper contains 4 sections, 11 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Explicit solution for viscous regularization
  3. Vanishing viscosity limit
  4. Solution of the inviscid equation \ref{['inviscid equation']}

Key Result

Theorem 2.1

The explicit solution $u^{\epsilon}, \epsilon>0$ of problem viscous equation is given by $u^{\epsilon}(x,t)=-\frac{2\epsilon \theta^{\epsilon}_x}{\theta^{\epsilon}},$ where and $R^{\epsilon},L^{\epsilon}$ are as follows: Here, the terms $\theta^{\epsilon}_0$ and $g$ are as follows: and $I_0$ is the Modified Bessel function of first kind. In addition, the term $f(t)$ is as follows: Further $u^{

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • proof
  • Lemma 4.1
  • ...and 10 more