Uniqueness and stability of steady-state solution with finite energy to the fractal Burgers equation

TL;DR

This paper investigates the steady-state solutions of the fractal Burgers equation with fractional dissipation, proving their existence, uniqueness, finite energy, and nonlinear stability on the real line.

Contribution

It establishes the existence and uniqueness of finite energy steady-state solutions and demonstrates their nonlinear stability, advancing understanding of fractional dissipative systems.

Findings

Existence of viscosity weak solutions to the fractal Burgers equation.

Uniqueness of steady-state solutions with finite $H^{rac{eta}{2}}$ energy.

Steady-state solutions are nonlinearly stable.

Abstract

The paper is concerned with the steady-state Burgers equation of fractional dissipation on the real line. We first prove the global existence of viscosity weak solutions to the fractal Burgers equation driven by the external force. Then the existence and uniqueness of solution with finite $H^{\frac{\alpha}{2}}$ energy to the steady-state equation are established by estimating the decay of fractal Burgers' solutions. Furthermore, we show that the unique steady-state solution is nonlinearly stable, which means any viscosity weak solution of fractal Burgers equation, starting close to the steady-state solution, will return to the steady state as $t\rightarrow\infty$.