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Hyperbolic systems of conservation laws.

Alberto Bressan (1999)

Revista Matemática Complutense

This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions....

Introduction to the models of phase transitions

A. Visintin (1998)

Bollettino dell'Unione Matematica Italiana

Le transizioni di fase si presentano in svariati processi fisici: un esempio tipico è la transizione solido-liquido. Il classico modello matematico, noto come problema di Stefan, tiene conto solo dello scambio del calore latente e della diffusione termica nelle fasi. Si tratta di un problema di frontiera libera, poiché l'evoluzione dell'interfaccia solido liquido è una delle incognite. In questo articolo si rivedono le formulazioni forte e debole di tale problema, e quindi si considerano alcune...

Inverse Problem for Fractional Diffusion Equation

Tuan, Vu Kim (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 33E12, 34K29, 34L15, 35K57, 35R30We prove that by taking suitable initial distributions only finitely many measurements on the boundary are required to recover uniquely the diffusion coefficient of a one dimensional fractional diffusion equation. If a lower bound on the diffusion coefficient is known a priori then even only two measurements are sufficient. The technique is based on possibility of extracting the full boundary spectral data from special lateral measurements.

Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

Luchko, Yury (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional...

Monotone operators. A survey directed to applications to differential equations

Jan Franců (1990)

Aplikace matematiky

The paper deals with the existence of solutions of the form A u = b with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions S and M . The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition ( M ) 0 . The second part which has a character of a survey compares various types of...

Nonlinear boundary value problems involving the extrinsic mean curvature operator

Jean Mawhin (2014)

Mathematica Bohemica

The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type ∇ · ∇ v 1 - | ∇ v | 2 = f ( | x | , v ) in B R , u = 0 on ∂ B R , where B R is the open ball of center 0 and radius R in ℝ n , and f is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.

On the Plasma-Charge problem

Mario Pulvirenti (2009/2010)

Séminaire Équations aux dérivées partielles

This short report is a review on recent results of S. Caprino, C. Marchioro, E. Miot and the author on the initial value problem associated to the evolution of a continuous distribution of charges (plasma) in presence of a finite number of point charges.

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...

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