Speaker
Description
We consider the dissipative magnetic Zakharov system in a smooth (2D) bounded domain
$\Omega\subset\mathbb{R}^2$ of the form
$ i E_t + \Delta E - n E + i E\times B +i\gamma_1 E = g_1(x,t), \qquad\qquad x\in \Omega, $
$ n_{tt}+\gamma_2 n_t -\Delta\left(n+|E|^2\right)=g_2(x,t), \qquad\qquad\qquad \qquad x\in \Omega, $
$ B_{tt}-\gamma_3\Delta B_t +\Delta^2\left(B+i E\times \overline{E}\right)=g_3(x,t), \qquad \qquad x\in \Omega, $
where $n(x,t)$ and $B(t, x) = (0, 0, B_3(t, x))$ are the real functions
and $E(x,t)=(E_1(t, x), E_2(t, x), 0)$ is a complex one.
If we omit magnetic field $B$, then the above system is reduced to the dissipative
Zakharov system. This system has been studied by many authors (see [1] and
references therein).
In the case $\Omega=\mathbb{R}^d$ for $d=2,3$ the Cauchy problem for the above system
has been considered in [2]. It was obtained local existence and uniqueness
results. Our main result is the global well-posedness
of the considered problem in some Sobolev type classes and existence of a global attractor.
[1] I. Chueshov and A. Shcherbina, On 2D Zakharov system in a bounded domain, Differential and Integral Equations, 18 (2005), 781-812.
[2] Boling Guo, Jingjun Zhang, Chunxiao Guo. \/, On the Cauchy problem for the magnetic Zakharov system, Monatshefte fur Mathematik, 170 (2013), 89-111.
