A geometric one-sided inequality for zero-viscosity limits

TL;DR

This paper unifies and generalizes one-sided inequalities like Oleinik's and Aronson-Benilan's for a broad class of PDEs, using a geometric approach based on the connectedness of certain level sets, leading to results on uniqueness and regularity.

Contribution

It introduces a geometric unification of one-sided inequalities for various PDEs, extending their applicability and deriving key properties without convexity assumptions.

Findings

Connectedness of level sets implies solution uniqueness.

Generalized inequalities apply to a wide class of first and second order equations.

Extended to multi-dimensional heat equation.

Abstract

The Oleinik inequality for conservation laws and Aronson-Benilan type inequalities for porous medium or p-Laplacian equations are one-sided inequalities that provide the fundamental features of the solution such as the uniqueness and sharp regularity. In this paper such one-sided inequalities are unified and generalized for a wide class of first and second order equations in the form of $$ u_t=\sigma(t,u,u_x,u_{xx}),\quad u(x,0)=u^0(x)\ge0,\quad t>0,\,x\in\bfR, $$ where the non-strict parabolicity ${\partial\over\partial q} \sigma(t,z,p,q)\ge0$ is assumed. The generalization or unification of one-sided inequalities is given in a geometric statement that the zero level set $$ A(t;m,x_0):=\{x:\rhom(x-x_0,t)-u(x,t)>0\} $$ is connected for all $t,m>0$ and $x_0\in\bfR$, where $\rhom$ is the fundamental solution with mass $m>0$. This geometric statement is shown to be equivalent to the previously mentioned one-sided inequalities and used to obtain uniqueness and TV boundedness of conservation laws without convexity assumption. Multi-dimensional extension for the heat equation is also given.