**Bernard Nowakowski**

**Afilliation:**Institute of Mathematics of the Polish Academy of Science

**Title:**Large time existence of strong solutions to MHD equations in cylindrical domains

**Abstract:**We investigate the existence of strong solutions to magnetohydrodynamics equations in bounded cylindrical domains in $\mathbb{R}^3$ under the Navier boundary conditions. These equations describe the interaction between a motion of a conductive fluid and a magnetic field. Assuming a sufficiently small rate of change of the external and the initial data along the axis of the cylinder we prove that strong solutions exist for any $T < \infty$ and that the velocity and the magnetic fields belong to $W^{2,1}_2(\Omega\times(0,T))$ whereas the gradient of the pressure to $L_2(\Omega\times(0,T))$.