By using the spectral Galerkin method, we prove the existence of weak solutions for a system of equations of magnetohydrodynamic type in non-cylindrical domains.

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium...

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution. ...

We study boundary layer solutions of the isentropic, compressible Navier-Stokes equations with Navier-friction boundary conditions when the viscosity constants appearing in the momentum equation are proportional to a small parameter $ϵ$. These boundary conditions are characteristic for the underlying inviscid problem, the compressible Euler equations.The boundary condition implies that the velocity on the boundary is proportional to the tangential component of the stress. The normal component of velocity...

This paper is devoted to the study of smooth flows of density-dependent fluids in ${ℝ}^N$ or in the torus ${𝕋}^N$. We aim at extending several classical results for the standard Euler or Navier-Stokes equations, to this new framework.Existence and uniqueness is stated on a time interval independent of the viscosity $\mu$ when $\mu$ goes to $0$. A blow-up criterion involving the norm of vorticity in $L^1(0,T;L^{\infty })$ is also proved. Besides, we show that if the density-dependent Euler equations have a smooth solution on a given time...

In questo lavoro si ricavano: 1) l'equazione d'onda linearizzata, 2) la formulazione Lagrangiana, 3) la formulazione Hamiltoniana, nella teoria della propagazione ondosa in un fluido comprimibile descritto dalle equazioni della magnetofluidodinamica ideale in presenza di corrente Hall.

In questo lavoro si ricavano: 1) l’equazione d’onda linearizzata, 2) la formulazione Lagrangiana, 3) la formulazione Hamiltoniana, nella teoria della propagazione ondosa in un fluido incomprimibile descritto dalle equazioni della magnetofluidodinamica ideale in presenza di corrente Hall.

We formulate the Leray problem for inhomogeneous fluids in two dimensions and outline the proof of the existence of a solution. There are two kinds of results depending on whether the given value for the density is a continuous function or only an $L^{\infty }$ function. In the former case, the given densities are attained in the sense of uniform convergence and in the latter with respect to weak-* convergence.

The mathematical theory of the passage from compressible to incompressible fluid flow is reviewed.