Three-dimensional anisotropic magneto-hydrodynamical system is investigated in the whole space ${ℝ}^3$. Existence and uniqueness results are proved in the anisotropic Sobolev space $H^{0,s}$ for $s\>1/2$. Asymptotic behavior of the solution when the Rossby number goes to zero is studied. The proofs, where the incompressibility condition is crucial, use the energy method, an appropriate dyadic decomposition of the frequency space, product laws in anisotropic Sobolev spaces and Strichartz-type estimates. ...

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...

In this work we study the existence, uniqueness and decay of solutions to a class of viscoelastic equations in a separable Hilbert space $H$given by $$\begin{array}{c}{u}_{tt}+M\left(\left[u\right]\right)Au-{\int }_0^{t}g(t-\tau )N\left(\left[u\right]\right)Aud\tau =0,\text{in}{L}^2(0,T;H)\\ u\left(0\right)=u_0,u_t\left(0\right)=u_1\end{array}$$ where by$\left[u\left(t\right)\right]$we are denoting $$\left[u\left(t\right)\right]=\left((u\left(t\right),u_t\left(t\right),(Au\left(t\right),u_t\left(t\right)),\parallel A^{\frac{1}{2}}u\left(t\right){\parallel }^2,\parallel A^{\frac{1}{2}}{u}_t\left(t\right){\parallel }^2,\parallel Au\left(t\right){\parallel }^2\right)\in {ℝ}^5$$ $A:D\left(A\right)\subset H\to H$ is a nonnegative, self-adjoint operator, $M$, $N:{\mathbb{R}}^5\to \mathbb{R}$ are $C^2$- functions and $g:\mathbb{R}\to \mathbb{R}$ is a $C^3$-function with appropriates conditions. We show that there exists global solution in time for small initial data. When $\left[u\left(t\right)\right]={∥A^{\frac{1}{2}}u∥}^2$ and $N=1$, we show the global existence for large initial data $\left(u_0,u_1\right)$ taken in the space $D\left(A\right)D\left(A^{1/2}\right)$ provided they are...