Solution properties of a 3D stochastic Euler fluid equation

We prove local well-posedness in regular spaces and a Beale-Kato-Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton's 2nd Law in every Lagrangian domain.Comment: Final version! Comments still welcome! Send email

On the existence and smoothness problem of the magnetohydrodynamics system

Fluid mechanics plays a pivotal role in engineering application to daily lives. The prominently famous fluid dynamics partial differential equations (PDE) due to its remarkable utility is the Navier-Stokes equations of which its mathematical and physical significance is so highly regarded that it has become one of the seven Millennium Prize problems declared by the Clay Research Institute. We study closely related systems of partial differential equations with focus on the magnetohydrodynamics system, of which its special case is the Navier-Stokes equations. Other systems of PDEs of our concern include the surface quasi-geostrophic equations, incompressible porous media equation governed by Darcy's law, Boussinesq system, Leray, Lans-alpha models, micropolar and magneto-micropolar fluid models. We discuss the properties of solutions to these systems such as the global regularity issue with fractional Laplacians, logarithmic supercriticality, component reduction results of regularity criteria

Remarks on Pressure Blow-Up Criterion of the 3D Zero-Diffusion Boussinesq Equations in Margin Besov Spaces

This study is focused on the pressure blow-up criterion for a smooth solution of three-dimensional zero-diffusion Boussinesq equations. With the aid of Littlewood-Paley decomposition together with the energy methods, it is proved that if the pressure satisfies the following condition on margin Besov spaces, π(x,t)∈L2/(2+r)(0,T;B˙∞,∞r) for r=±1, then the smooth solution can be continually extended to the interval (0,T⁎) for some T⁎>T. The findings extend largely the previous results

Navier-Stokes Equations with Navier Boundary Conditions and Stochastic Lie Transport: Well-Posedness and Inviscid Limit

We prove the existence and uniqueness of global, probabilistically strong, analytically strong solutions of the 2D Stochastic Navier-Stokes Equation under Navier boundary conditions. The choice of noise includes a large class of additive, multiplicative and transport models. We emphasise that with a transport type noise, the Navier boundary conditions enable direct energy estimates which appear to be prohibited for the usual no-slip condition. The importance of the Stochastic Advection by Lie Transport (SALT) structure, in comparison to a purely transport Stratonovich noise, is also highlighted in these estimates. In the particular cases of SALT noise, the free boundary condition and a domain of non-negative curvature, the inviscid limit exists and is a global, probabilistically weak, analytically weak solution of the corresponding Stochastic Euler Equation