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 $L^\infty$-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