A geometric one-sided inequality for zero-viscosity limits

The Oleinik inequality for conservation laws and AronsonBenilan 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 ut = σ(t, u, ux, uxx), u(x, 0) = u (x) ≥ 0, t > 0, x ∈ R, where the non-strict parabolicity ∂ ∂qσ(t, z, p, q) ≥ 0 is assumed. The generalization or unification of one-sided inequalities is given in a geometric statement that the zero level set A(t;m,x0) := {x : ρm(x− x0, t)− u(x, t) > 0} is connected for all t,m > 0 and x0 ∈ R, where ρm 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.