Parabolic Kirchhoff equations with non-homogeneous flux boundary conditions: well-posedness, regularity and asymptotic behavior

Nonlinearity ◽  
2021 ◽  
Vol 34 (8) ◽  
pp. 5844-5871
Author(s):  
Tito L Mamani Luna ◽  
Gustavo Ferron Madeira
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Kirchhoff Equations ◽  
Flux Boundary Conditions

MATHEMATICAL STUDY OF A HYPERBOLIC REGULARIZATION TO ENSURE GAUSS' LAW CONSERVATION IN MAXWELL–VLASOV APPLICATIONS

2012 ◽  
Vol 22 (04) ◽  
pp. 1150020 ◽  
Author(s):  
BRUNO FORNET ◽  
VINCENT MOUYSSET ◽  
ÁNGEL RODRÍGUEZ-ARÓS
Keyword(s):  
Numerical Simulation ◽  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Plasma Physics ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Well Posedness ◽  
Mathematical Study ◽  
Particle In Cell ◽  
Parameter Dependent

This paper studies a hyperbolic modification of Maxwell's equations to ensure Gauss' law. This correction was obtained by adding a parameter-dependent new unknown and is of great interest for the numerical simulation in plasma physics since the discretization of the Maxwell–Vlasov system does not grant straightforwardly the physical conservation of the charge. Such problems are encountered while using Particle-In-Cell schemes. In this paper the new proposed system has the interest of still being a Friedrichs' one. Its asymptotic behavior with respect to the parameter and the link between modified and original Maxwell's systems are thus investigated. At last, we look for some boundary conditions, granting the well-posedness of the system. Generalizations of the Silver–Müller condition, perfect electric and magnetic conductors, as well as impedance and admittance representation of materials are detailed.

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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