2018/12/05 by Nathaël Alibaud, Alibaud, Nathaël, Jørgen Endal +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1812.02058
openalex publication_date 2018/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality: ∫ |St u0-St v0| φ0 dx≤ ∫ |u0-v0| Gt φ0 dx, ∀ φ0 ≥ 0, ∀ u0, ∀ v0, (⋆) where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation ∂t u+div F(u) = div (A(u) D u), and where we look for the smallest semigroup Gt satisfying (⋆). This amounts to finding an optimal weighted L1 contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation∂t φ= supξ\F'(ξ) ⋅ D φ+tr(A(ξ) D2φ)\. Since weighted L1 contraction results are mainly used for possibly nonintegrable L^∞ solutions u, the natural spaces behind this duality are L^∞ for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions φ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it -- a subspace of L1 called L^∞int. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].