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An existence and uniqueness result using bounded variation estimates in Galerkin approximations

Ramesh Mondal, Aditi Sengupta

TL;DR

This work proves the existence and uniqueness of weak solutions for a class of quasilinear parabolic initial-boundary value problems by establishing bounded variation (BV) estimates for Galerkin approximations. The main novelty is applying BV-compactness to extract an almost everywhere convergent subsequence from Galerkin schemes, using a Bardos-et-al style approach and functional-analytic tools (Hahn–Banach, Riesz) to obtain $L^1(\Omega_T)$ control of time derivatives. The limit of the BV-bounded Galerkin sequence is shown to be a weak solution, with uniqueness derived from standard linear parabolic theory. Overall, the paper provides an alternative BV-based route to existence (and standard uniqueness) for quasilinear parabolic problems and highlights the efficiency of BV-compactness in the Galerkin context.

Abstract

Bounded variation estimates of Galerkin approximations are established in order to extract an almost everywhere convergent subsequence of Galerkin approximations. As a result we prove existence of weak solutions of initial boundary value problems for quasilinear parabolic equations. Uniqueness of weak solutions is derieved applying a standard argument.

An existence and uniqueness result using bounded variation estimates in Galerkin approximations

TL;DR

This work proves the existence and uniqueness of weak solutions for a class of quasilinear parabolic initial-boundary value problems by establishing bounded variation (BV) estimates for Galerkin approximations. The main novelty is applying BV-compactness to extract an almost everywhere convergent subsequence from Galerkin schemes, using a Bardos-et-al style approach and functional-analytic tools (Hahn–Banach, Riesz) to obtain control of time derivatives. The limit of the BV-bounded Galerkin sequence is shown to be a weak solution, with uniqueness derived from standard linear parabolic theory. Overall, the paper provides an alternative BV-based route to existence (and standard uniqueness) for quasilinear parabolic problems and highlights the efficiency of BV-compactness in the Galerkin context.

Abstract

Bounded variation estimates of Galerkin approximations are established in order to extract an almost everywhere convergent subsequence of Galerkin approximations. As a result we prove existence of weak solutions of initial boundary value problems for quasilinear parabolic equations. Uniqueness of weak solutions is derieved applying a standard argument.
Paper Structure (6 sections, 7 theorems, 102 equations)

This paper contains 6 sections, 7 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Existence of solution
  3. Existence of Galerkin approximations
  4. Energy estimates
  5. BV-Estimate
  6. Convergence

Key Result

Theorem 1.1

Let $f$, $\left(B_{ij}\right)$ and $u_{0}$ satisfy Hypothesis-H1. Then there exists a unique solution $u$ in $W(0,T)$ of the initial boundary value problem eq1.1, eq1.2, eq1.3 which satisfies Aditi_RM_equation1_Defn and Aditi_RM_equation2_Defn.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 1 more