A geometric one-sided inequality for zero-viscosity limits

الأصل · EN

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 uₜ=σ(t,u,uₓ,uxx), u(x,0)=u⁰(x)≥0, t>0,x∈, where the non-strict parabolicity ∂∂ q σ(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:(x-x₀,t)-u(x,t)>0} is connected for all t,m>0 and x₀∈, where 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.