We prove existence of weak solutions of stochastic Navier-Stokes equations in R which do not satisfy the coercivity condition. The equations are formally derived from the critical point of some variational problem defined on the space of volume preserving diffeomorphisms in R. Since the domain of our equation is unbounded, it is more difficult to get tightness of approximating sequences of solu...

We establish a Navier–Stokes–Fourier limit for solutions of the Boltzmann equation considered over any periodic spatial domain of dimension two or more. We do this for a broad class of collision kernels that relaxes the Grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials. Appropriately scaled families of DiPerna–Lions renormaliz...

We are interested in the rate of convergence of solutions of 2D Navier-Stokes equations in a smooth bounded domain as the viscosity tends to zero under Navier friction condition. If the initial velocity is smooth enough( ), it is known that the rate of convergence is linearly propotional to the viscosity. Here, we consider the rate of convergence for nonsmooth velocity fields when the gradient ...

Here u, b describe the flow velocity vector and the magnetic field vector respectively, p is a scalar pressure, ν > 0 is the kinematic viscosity, η > 0 is the magnetic diffusivity, while u0 and b0 are the given initial velocity and initial magnetic field with ∇ · u0 = ∇ · b0 = 0. If ν = η = 0, (1.1) is called the ideal MHD equations. As same as the 3D Navier-Stokes equations, the regularity of ...

Two-level finite element methods are applied to solve numerically the 2D/3D steady Navier-Stokes equations if a strong uniqueness condition ( ‖f‖ −1 ‖f‖0 ) 1 2 ≤ δ = 1 − N‖f‖ −1 ν2 holds, where N is defined in (2.4)-(2.6). Moreover, one-level finite element method is applied to solve numerically the 2D/3D steady Navier-Stokes equations if a weak uniqueness condition 0 < δ < ( ‖f‖ −1 ‖f‖0 ) 1 2 ...

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ωε, whose thickness is of order O(ε) as ε → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ωε, and to the periodicity condition on its sides. Assume that the frictio...