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Towards a Schauder theory for fractional viscous Hamilton--Jacobi equations

Espen R. Jakobsen, Artur Rutkowski

Abstract

We survey some results on Lipschitz and Schauder regularity estimates for viscous Hamilton--Jacobi equations with subcritical Lévy diffusions. The Schauder estimates, along with existence of smooth solutions, are obtained with the help of a Duhamel formula and $L^1$ bounds on the spatial derivatives of the heat kernel. Our results cover very general nonlocal and mixed local-nonlocal diffusions, including strongly anisotropic, nonsymmetric, mixed order, and spectrally one-sided models.

Towards a Schauder theory for fractional viscous Hamilton--Jacobi equations

Abstract

We survey some results on Lipschitz and Schauder regularity estimates for viscous Hamilton--Jacobi equations with subcritical Lévy diffusions. The Schauder estimates, along with existence of smooth solutions, are obtained with the help of a Duhamel formula and bounds on the spatial derivatives of the heat kernel. Our results cover very general nonlocal and mixed local-nonlocal diffusions, including strongly anisotropic, nonsymmetric, mixed order, and spectrally one-sided models.
Paper Structure (5 sections, 5 theorems, 27 equations)

This paper contains 5 sections, 5 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Assumptions
  3. Comparison principle and Lipschitz estimates
  4. Uniform Schauder estimates with regular initial data
  5. Schauder estimates with irregular initial data

Key Result

Theorem 3.1

Assume that eq:H0, eq:H2, eq:H3, and eq:F0 hold. Let $u_0$ be uniformly continuous and bounded and suppose that $u$ and $v$ are respectively classical sub- and supersolution to eq:HJ, i.e., $u(0,x)\leq u_0(x)\leq v(0,x)$ for $x\in \mathbb{R}^d$ and Then $u\leq v$ on $[0,T)\times \mathbb{R}^d$.

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • Remark 3.4
  • proof : Proof of Theorem \ref{['th:Lip']} for regular data and solutions
  • Theorem 4.1
  • proof
  • ...and 1 more