# Regularizing effect of homogeneous evolution equations: case homogeneous order zero

@article{Hauer2019RegularizingEO, title={Regularizing effect of homogeneous evolution equations: case homogeneous order zero}, author={Daniel Hauer and Jos{\'e} M. Maz{\'o}n}, journal={arXiv: Analysis of PDEs}, year={2019} }

In this paper, we develop a functional analytical theory for establishing that mild solutions of first-order Cauchy problems involving homogeneous operators of order zero are strong solutions; in particular, the first-order time derivative satisfies a global regularity estimate depending only on the initial value and the positive time. We apply those results to the Cauchy problem associated with the total variational flow operator and the nonlocal fractional 1-Laplace operator.

## References

SHOWING 1-10 OF 19 REFERENCES

### Regularizing Effects of Homogeneous Evolution Equations

- Mathematics
- 1980

Abstract : It is well-known that solving the initial-value problem for the heat equation forward in time takes a 'rough' initial temperature into a temperature which is smooth at later times t >…

### Chapter 1 Semigroups and evolution equations: Functional calculus, regularity and kernel estimates

- Mathematics
- 2002

### Vector-valued Laplace Transforms and Cauchy Problems

- Mathematics
- 2011

This monograph gives a systematic account of the theory of vector-valued Laplace transforms, ranging from representation theory to Tauberian theorems. In parallel, the theory of linear Cauchy…

### Nonlinear Differential Equations of Monotone Types in Banach Spaces

- Mathematics
- 2010

Fundamental Functional Analysis.- Maximal Monotone Operators in Banach Spaces.- Accretive Nonlinear Operators in Banach Spaces.- The Cauchy Problem in Banach Spaces.- Existence Theory of Nonlinear…

### Analysis and Geometry on Groups

- Mathematics
- 1993

Preface Foreword 1. Introduction 2. Dimensional inequalities for semigroups of operators on the Lp spaces 3. Systems of vector fields satisfying Hormander's condition 4. The heat kernel on nilpotent…

### Parabolic Quasilinear Equations Min-imizing Linear Growth Functionals

- Mathematics
- 2004

1 Total Variation Based Image Restoration.- 1.1 Introduction.- 1.2 Equivalence between Constrained and Unconstrained Restoration.- 1.3 The Partial Differential Equation Satisfied by the Minimum of…

### Heat Kernel and Analysis on Manifolds

- Mathematics
- 2012

Laplace operator and the heat equation in $\mathbb{R}^n$ Function spaces in $\mathbb{R}^n$ Laplace operator on a Riemannian manifold Laplace operator and heat equation in $L^{2}(M)$ Weak maximum…

### A First Course in Sobolev Spaces

- Mathematics
- 2009

Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural…