Existence and Uniqueness of Mild Solutions of Viscous Burgers' Equation With Dirichlet Boundary Conditions

Abstract

ABSTRACT In this paper, we solve for the initial‐boundary value problem involving the viscous Burgers equation with a homogeneous Dirichlet boundary and the given initial condition . We investigate the existence of mild solutions in the Banach space , for a sufficiently small time horizon . The analysis relies on semigroup theory and the formulation of the problem as a fixed‐point equation involving an integral operator. We prove that the associated nonlinear map is compact and continuous, and satisfies suitable a priori estimates. The Leray–Schauder topological degree is then applied to establish the existence of a fixed point, which corresponds to a mild solution of the original problem. Our result confirms local‐in‐time solvability under minimal regularity assumptions on the initial data.