Conditional Regularity of Weak Solutions to the 3D Magnetic Bénard Fluid System

2021 ◽  
Vol 34 (2) ◽  
pp. 144-169
Author(s):  
Ma Liangliang
Keyword(s):  
Weak Solutions ◽  
Fluid System ◽  
Regularity Of Weak Solutions

Regularity of Weak Solutions to a Class of Nonlinear Problem

2021 ◽  
Vol 41 (4) ◽  
pp. 1333-1365
Author(s):  
Jianfeng Zhou ◽  
Zhong Tan
Keyword(s):  
Nonlinear Problem ◽  
Weak Solutions ◽  
Regularity Of Weak Solutions

Regularity of weak solutions for the relativistic Vlasov–Maxwell system

2018 ◽  
Vol 15 (04) ◽  
pp. 693-719 ◽  
Author(s):  
Nicolas Besse ◽  
Philippe Bechouche
Keyword(s):  
Electromagnetic Field ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Regularity Of Solutions ◽  
Maxwell System ◽  
Smoothing Effect ◽  
Regular Solutions ◽  
Low Velocity ◽  
Regularity Of Weak Solutions ◽  
The One

We investigate the regularity of weak solutions of the relativistic Vlasov–Maxwell system by using Fourier analysis and the smoothing effect of low velocity particles. This smoothing effect has been used by several authors (see Glassey and Strauss 1986; Klainerman and Staffilani, 2002) for proving existence and uniqueness of [Formula: see text]-regular solutions of the Vlasov–Maxwell system. This smoothing mechanism has also been used to study the regularity of solutions for a kinetic transport equation coupled with a wave equation (see Bouchut, Golse and Pallard 2004). Under the same assumptions as in the paper “Nonresonant smoothing for coupled wave[Formula: see text]+[Formula: see text]transport equations and the Vlasov–Maxwell system”, Rev. Mat. Iberoamericana 20 (2004) 865–892, by Bouchut, Golse and Pallard, we prove a slightly better regularity for the electromagnetic field than the one showed in the latter paper. Namely, we prove that the electromagnetic field belongs to [Formula: see text], with [Formula: see text].

Strict convexity, strong ellipticity, and regularity in the calculus of variations

1980 ◽  
Vol 87 (3) ◽  
pp. 501-513 ◽  
Author(s):  
J. M. Ball
Keyword(s):  
Nonlinear Systems ◽  
Calculus Of Variations ◽  
Weak Solutions ◽  
Convex Functions ◽  
Strict Convexity ◽  
Mathematical Tool ◽  
Strong Ellipticity ◽  
Regularity Of Weak Solutions ◽  
Nonlinear Elastostatics

In this paper we investigate the connection between strong ellipticity and the regularity of weak solutions to the equations of nonlinear elastostatics and other nonlinear systems arising from the calculus of variations. The main mathematical tool is a new characterization of continuously differentiable strictly convex functions. We first describe this characterization, and then explain how it can be applied to the calculus of variations and to elastostatics.

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