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Decomposition and Moser's lemma.

David E. Edmunds, Miroslav Krbec (2002)

Revista Matemática Complutense

Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...

Décomposition formelle d'un système microdifférentiel aux points génériques

Rui Rodrigues (1992)

Annales de l'institut Fourier

Soit X une variété analytique complexe et T * X → X son fibre cotangent. Soit M un module cohérent sur l’anneau des opérateurs microdifférentiels formels sur X . Dans le cas ou le support (ou variété caractéristique) de M est une hypersurface, B. Malgrange a démontre que M se décompose en systèmes élémentaires au point générique et après tensorisation par l’anneau des opérateurs microdifférentiels d’ordre q - fractionnaire avec q approprie.Dans ce travail, on généralise le résultat cité : d’abord pour un...

Deep learning for gradient flows using the Brezis–Ekeland principle

Laura Carini, Max Jensen, Robert Nürnberg (2023)

Archivum Mathematicum

We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis–Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited for a machine learning approach using deep neural networks. We describe our approach in a general framework and illustrate the method with the help of an example implementation for the heat equation in space dimensions two to seven.

Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes

Christian Heinemann, Christiane Kraus (2014)

Mathematica Bohemica

This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field. The analysis is performed on a time-dependent...

Description of the lack of compactness of some critical Sobolev embedding

Hajer Bahouri (2011)

Journées Équations aux dérivées partielles

In this text, we present two recent results on the characterization of the lack of compactness of some critical Sobolev embedding. The first one derived in [5] deals with an abstract framework including Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. The second one established in [3] concerns the lack of compactness of H 1 ( ℝ 2 ) into the Orlicz space. Although the two results are expressed in the same manner (by means of defect measures) and rely on the defect of compactness due to concentration...

Development of small and large compressive pulses in two-phase flow

Nishi Deepa Palo, Jasobanta Jena, Meera Chadha (2024)

Applications of Mathematics

The evolutions of small and large compressive pulses are studied in a two-phase flow of gas and dust particles with a variable azimuthal velocity. The method of relatively undistorted waves is used to study the mechanical pulses of different types in a rotational, axisymmetric dusty gas. The results obtained are compared with that of nonrotating medium. Asymptotic expansion procedure is used to discuss the nonlinear theory of geometrical acoustics. The influence of the solid particles and the rotational...

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