12 research outputs found
On diffusive 2D Fokker-Planck-Navier-Stokes systems
We study models kinetic models of polymeric fluids. We introduce a notion of
solutions which is based on moments of polymeric distributions. We prove global
existence and uniqueness of a large class of initial data for diffusive systems
of kinetic equations coupled to fluid equations. As a corollary, we obtain a
rigorous derivation of Oldroyd-B closure. We also prove decay of free energy
for all the systems considered
Local Well-posedness for the Kinetic MMT Model
The MMT equation was proposed by Majda, McLaughlin and Tabak as a model to
study wave turbulence. We focus on the kinetic equation associated to this
Hamiltonian system, which is believed to give a way to predict turbulent
spectra. We clarify the formulation of the problem, and we develop the local
well-posedness theory for this equation. Our analysis uncovers a surprising
nonlinear smoothing phenomenon
A uniform bound for solutions to a thermo-diffusive system
We obtain uniform in time -bounds for the solutions to a class of
thermo-diffusive systems. The nonlinearity is assumed to be at most
sub-exponentially growing at infinity and have a linear behavior near zero.Comment: arXiv admin note: text overlap with arXiv:2204.1035
On the Models of the Fluid-Polymer Systems
The purpose of this work is to study fluid-polymer systems. A fluid-polymer system is a system consisting of solvent fluids and polymers, either suspended in the bulk (polymeric fluid systems) or attached on the boundaries. Mathematically, they are coupled multi-scale systems of partial differential equations, consisting of a fluid portion modeled by the Navier-Stokes equation, and a polymer portion modeled by the Fokker-Planck equation. Key difficulties lie in the coupling of two equations.
We propose a new approach to show the well-posedness of a certain class of polymeric fluid systems. In this approach, we use ``moments" to translate a multi-scale system to a fully macroscopic system (consisting of infinitely many equations), solve the macroscopic system, and recover the solution of the original multi-scale system. As an application, we obtain the large data global well-posedness of a certain class of polymeric fluid systems.
We also show the local well-posedness when a polymeric fluid system is written in Lagrangian coordinates. This approach allows us to show the uniqueness in lower regularity space and the Lipschitz dependence on initial data.
Finally, we propose a new boundary condition which describes the situation where polymers are attached on the fluid-wall interface. Using kinetic theory, we derive a dynamic boundary condition which can be interpreted as a ``history-dependent slip" boundary condition, and we prove global well-posedness in 2D case. Also, we show that the inviscid limit holds for an incompressible Navier-Stokes system with this boundary condition
Invariant measures for stochastic conservation laws on the line
We consider a stochastic conservation law on the line with solution-dependent
diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth,
spatially-homogeneous kick-type random forcing. We show that this Markov
process admits a unique ergodic spatially-homogeneous invariant measure for
each mean in a non-explicit unbounded set. This generalizes previous work on
the stochastic Burgers equation.Comment: 33 pages; generalized assumptions on the noise in this versio