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Régularité de la solution d'un problème de Cauchy fortement non linéaire à données singulières en un point

Jean-Yves Chemin (1989)

Annales de l'institut Fourier

Dans cet article, on étudie la régularité d’une solution réelle, appartenant à H s pour s assez grand, d’une équation aux dérivées partielles strictement hyperbolique et fortement non linéaire d’ordre deux. On suppose que les données de Cauchy sur une hypersurface spatiale lisse sont régulières en dehors d’un point, et ont une singularité conormale en ce point; on démontre alors que la réunion Γ des bicaractéristiques nulles issues de ce point est, en dehors de ce point, une hypersurface lisse et...

Regularity and Blow up for Active Scalars

A. Kiselev (2010)

Mathematical Modelling of Natural Phenomena

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow...

Regularity and optimal control of quasicoupled and coupled heating processes

Jiří Jarušek (1996)

Applications of Mathematics

Sufficient conditions for the stresses in the threedimensional linearized coupled thermoelastic system including viscoelasticity to be continuous and bounded are derived and optimization of heating processes described by quasicoupled or partially linearized coupled thermoelastic systems with constraints on stresses is treated. Due to the consideration of heating regimes being “as nonregular as possible” and because of the well-known lack of results concerning the classical regularity of solutions...

Regularity of Lipschitz free boundaries for the thin one-phase problem

Daniela De Silva, Ovidiu Savin (2015)

Journal of the European Mathematical Society

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional E ( u , Ω ) = ∫ Ω | ∇ u | 2 d X + ℋ n ( { u > 0 } ∩ { x n + 1 = 0 } ) , Ω ⊂ ℝ n + 1 , among all functions u ≥ 0 which are fixed on ∂ Ω .

Regularity of optimal shapes for the Dirichlet’s energy with volume constraint

Tanguy Briancon (2004)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove some regularity results for the boundary of an open subset of ℝ d which minimizes the Dirichlet’s energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative.

Regularity of optimal shapes for the Dirichlet's energy with volume constraint

Tanguy Briancon (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove some regularity results for the boundary of an open subset of d which minimizes the Dirichlet's energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative.

Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u t = ∇ · ( u ∇ ( - Δ ) - s u ) , 0 < s < 1 . The problem is posed in { x ∈ ℝ n , t ∈ ℝ } with nonnegative initial data u ( x , 0 ) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and C α regularity of such weak solutions. Finally, we extend the existence...

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