On the energy equality for very weak solutions to 3D MHD equations

2021 ◽  
pp. 1-24
Author(s):  
Baishun Lai ◽  
Yifan Yang
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Weak Solutions ◽  
Mhd Equations ◽  
Time Interval ◽  
Very Weak Solutions ◽  
Energy Equality ◽  
Very Weak Solution

In this paper, we consider the energy equality of the 3D Cauchy problem for the magneto-hydrodynamics (MHD) equations. We show that if a very weak solution of MHD equations belongs to $L^{4}(0,\,T;L^{4}(\mathbb {R}^{3}))$ , then it is actually in the Leray–Hopf class and therefore must satisfy the energy equality in the time interval $[0,\,T]$ .

scholarly journals Integrability of Very Weak Solutions for Boundary Value Problems of Nonhomogeneous p-Harmonic Equations

2019 ◽  
pp. 1-9
Author(s):  
Yeqing Zhu ◽  
Yanxia Zhou ◽  
Yuxia Tong
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Boundary Value ◽  
Very Weak Solutions ◽  
Very Weak Solution ◽  
Harmonic Equation

The paper deals with very weak solutions u to boundary value problems of the nonhomogeneous p-harmonic equation. We show that, any very weak solution u to the boundary value problem is integrable provided that r is sufficiently close to p.

REGULARITY OF VERY WEAK SOLUTIONS FOR NONHOMOGENEOUS ELLIPTIC EQUATION

2013 ◽  
Vol 15 (04) ◽  
pp. 1350012 ◽  
Author(s):  
WEI ZHANG ◽  
JIGUANG BAO
Keyword(s):  
Weak Solution ◽  
Elliptic Equation ◽  
Weak Solutions ◽  
Local Regularity ◽  
Difference Quotient ◽  
Very Weak Solutions ◽  
Very Weak Solution ◽  
Bootstrap Argument ◽  
The Difference ◽  
Higher Regularity

In this paper, we study the local regularity of very weak solution [Formula: see text] of the elliptic equation Dj(aij(x)Diu) = f - Digi. Using the bootstrap argument and the difference quotient method, we obtain that if [Formula: see text], [Formula: see text] and [Formula: see text] with 1 < p < ∞, then [Formula: see text]. Furthermore, we consider the higher regularity of u.

Local Boundedness for Very Weak Solutions of Leray-Lions Equation

2012 ◽  
Vol 457-458 ◽  
pp. 210-213
Author(s):  
Jian Tao Gu ◽  
Chun Xia Gao ◽  
Yu Xia Tong
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Decomposition Methods ◽  
Hodge Decomposition ◽  
Very Weak Solutions ◽  
Local Boundedness ◽  
Very Weak Solution

The local boundedness of very weak solution of Leray-Lions equation is given in this paper by Hodge decomposition methods.

scholarly journals On the regularity of very weak solutions for linear elliptic equations in divergence form

2020 ◽  
Vol 27 (5) ◽  
Author(s):  
Domenico Angelo La Manna ◽  
Chiara Leone ◽  
Roberta Schiattarella
Keyword(s):  
Weak Solution ◽  
Elliptic Equation ◽  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Continuity Condition ◽  
Very Weak Solutions ◽  
Linear Elliptic Equation ◽  
Very Weak Solution

Abstract In this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .

Uniqueness for Very Weak Solution to a Class of Elliptic Equations

2012 ◽  
Vol 457-458 ◽  
pp. 863-866
Author(s):  
Xu Juan Xu ◽  
Jian Tao Gu ◽  
Xiao Li Liu
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Hodge Decomposition ◽  
Very Weak Solutions ◽  
Uniqueness Property ◽  
Very Weak Solution

This paper studies on the very weak solution to a class of elliptic equations , and acquire the uniqueness property for very weak solutions by means of the Hodge decomposition and others.

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

scholarly journals On the uniqueness of weak solutions to the Ericksen–Leslie liquid crystal model in ℝ2

2016 ◽  
Vol 26 (04) ◽  
pp. 803-822 ◽  
Author(s):  
Jinkai Li ◽  
Edriss S. Titi ◽  
Zhouping Xin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Liquid Crystal ◽  
Weak Solutions ◽  
Energy Estimates ◽  
Unique Weak Solution ◽  
The Cauchy Problem ◽  
Crystal Model ◽  
Main Ideas ◽  
Leslie System

This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen–Leslie system of liquid crystal models in [Formula: see text], with both general Leslie stress tensors and general Oseen–Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen–Leslie system.

scholarly journals A New Regularity Criterion for the Three-Dimensional Incompressible Magnetohydrodynamic Equations in the Besov Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
TianLi LI ◽  
Wen Wang ◽  
Lei Liu
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Besov Spaces ◽  
Pressure Field ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Mhd Equations ◽  
Regularity Criteria ◽  
Magnetohydrodynamic Equations ◽  
Incompressible Magnetohydrodynamic Equations

Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. Our results imply that the scalar pressure field π plays an important role in the regularity problem of MHD equations. We derive that the weak solution u , b is regular on 0 , T , which is provided for the scalar pressure field π in the Besov spaces.

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