scholarly journals Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations

2020 ◽  
Vol 19 (3) ◽  
pp. 1509-1535
Author(s):  
G. Deugoué ◽  
◽  
J. K. Djoko ◽  
A. C. Fouape ◽  
A. Ndongmo Ngana ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Magnetohydrodynamic Equations

A BLOW-UP CRITERION FOR 3D COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH VACUUM

2012 ◽  
Vol 22 (02) ◽  
pp. 1150010 ◽  
Author(s):  
XINYING XU ◽  
JIANWEN ZHANG
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Magnetohydrodynamic Equations ◽  
The Cauchy Problem ◽  
Blow Up Criterion

This paper is concerned with a blow-up criterion of strong solutions for three-dimensional compressible isentropic magnetohydrodynamic equations with vacuum. It is shown that if the density and velocity satisfy [Formula: see text], where [Formula: see text], 3 < r ≤ ∞ and [Formula: see text] denotes the weak Lr-space, then the strong solutions to the Cauchy problem of the compressible magnetohydrodynamic equations can exist globally over [0, T].

scholarly journals The Yang-Mills heat equation with finite action in three dimensions

2022 ◽  
Vol 275 (1349) ◽  
Author(s):  
Leonard Gross
Keyword(s):  
Heat Equation ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Three Dimensions ◽  
Dimensional Manifold ◽  
Uniqueness Of Solutions ◽  
Reconstruction Procedure ◽  
Content Type ◽  
Yang Mills ◽  
Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

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