instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
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Existence and uniqueness for the Ladyzhenskaya model of incompressible viscous fluid and one introduction of the pullback attractors
Seminario PHD
we establish a result of existence (global in time) and uniqueness of solution for the Ladyzhenskaya model of incompressible viscous fluid in a domain $\Omega\subset{\mathbb{R}}^{n}$, $n\geq2$. The motion of incompressible, viscous fluids in $\Omega$, characterized by the velocity field $u=(u_{1},...,u_{n})$ and the pressure $\pi$, is governed by the system of $n+1$ equations

 \begin{equation}
\left\{\begin{array}{l}
  \frac{\partial u}{\partial t}-div_{x}S(Du)+div_{x}(u\otimes u)+\nabla_{x}\pi=f\;\;in\;(\tau,+\infty)\times\Omega,\\
 div_{x}u=0\;\;in\;(\tau,+\infty)\times\Omega,\\
u(\tau,x)=u_{\tau}(x),\;\;x\in\Omega,\\
u=0\;\;on\;(\tau,+\infty)\times\partial\Omega,
\end{array}\right.
  \end{equation}
 where the operator $S$ is a potential. If we have time, we will make some comments on the long-time behavior.

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