On the uniqueness and stability of weak solutions of a fokker-planck-vlasov equation

1978 ◽  
pp. 141-150 ◽  
Author(s):  
Reimund Rautmann
Keyword(s):  
Weak Solutions ◽  
Vlasov Equation ◽  
Fokker Planck

Global weak solutions of the Navier–Stokes–Fokker–Planck system

2013 ◽  
Vol 65 (2) ◽  
pp. 212-248 ◽  
Author(s):  
S. M. Egorov ◽  
E. Ya. Khruslov
Keyword(s):  
Weak Solutions ◽  
Navier Stokes ◽  
Global Weak Solutions ◽  
Fokker Planck

GLOBAL WEAK SOLUTIONS FOR A VLASOV–FOKKER–PLANCK/NAVIER–STOKES SYSTEM OF EQUATIONS

2007 ◽  
Vol 17 (07) ◽  
pp. 1039-1063 ◽  
Author(s):  
A. MELLET ◽  
A. VASSEUR
Keyword(s):  
Weak Solutions ◽  
Stokes System ◽  
Fluid Velocity ◽  
Planck Equation ◽  
Coupled System ◽  
Navier Stokes ◽  
Global Weak Solutions ◽  
Navier Stokes Equation ◽  
Reflection Boundary ◽  
Fokker Planck

We establish the existence of a weak solutions for a coupled system of kinetic and fluid equations. More precisely, we consider a Vlasov–Fokker–Planck equation coupled to compressible Navier–Stokes equation via a drag force. The fluid is assumed to be barotropic with γ-pressure law (γ > 3/2). The existence of weak solutions is proved in a bounded domain of ℝ3 with homogeneous Dirichlet conditions on the fluid velocity field and Dirichlet or reflection boundary conditions on the kinetic distribution function.

scholarly journals Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fucai Li ◽  
Yue Li
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Local Alignment ◽  
Fokker Planck

<p style='text-indent:20px;'>We study a kinetic-fluid model in a <inline-formula><tex-math id="M1">\begin{document}$ 3D $\end{document}</tex-math></inline-formula> bounded domain. More precisely, this model is a coupling of the Vlasov-Fokker-Planck equation with the local alignment force and the compressible Navier-Stokes equations with nonhomogeneous Dirichlet boundary condition. We prove the global existence of weak solutions to it for the isentropic fluid (adiabatic coefficient <inline-formula><tex-math id="M2">\begin{document}$ \gamma&gt; 3/2 $\end{document}</tex-math></inline-formula>) and hence extend the existence result of Choi and Jung [Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain, arXiv: 1912.13134v2], where the velocity of the fluid is supplemented with homogeneous Dirichlet boundary condition.</p>

scholarly journals EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR SOME POLYMERIC FLOW MODELS

2005 ◽  
Vol 15 (06) ◽  
pp. 939-983 ◽  
Author(s):  
JOHN W. BARRETT ◽  
CHRISTOPH SCHWAB ◽  
ENDRE SÜLI
Keyword(s):  
Stress Tensor ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Polymer Chains ◽  
Elastic Model ◽  
Spring Model ◽  
Extra Stress Tensor ◽  
Spring Force ◽  
Extra Stress ◽  
Fokker Planck

We study the existence of global-in-time weak solutions to a coupled microscopic–macroscopic bead-spring model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2, 3, for the velocity and the pressure of the fluid, with an extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function which satisfies a Fokker–Planck type degenerate parabolic equation. Upon appropriate smoothing of the convective velocity field in the Fokker–Planck equation, and in some circumstances, of the extra-stress tensor, we establish the existence of global-in-time weak solutions to this regularised bead-spring model for a general class of spring-force-potentials including in particular the widely used FENE (Finitely Extensible Nonlinear Elastic) model.

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