Seminar on Partial Differential Equations

usually takes place each Tuesday at 09:00 in IM, rear building, ground floor
Chair: Šárka Nečasová, Milan Pokorný

Strong solutions to the steady compressible Navier-Stokes equations with inflow boundary conditions

Tomasz Piasecki

University of Warsaw

Tuesday, 2. October 2018 - 9:00 to 10:00

We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition in a vicinity of given laminar solutions under the assumption that the pressure is a linear function of the density. In particular, we do not require any information on the gradient of the density or second gradient of the velocity. Our result holds provided certain condition on the shape of the boundary around the points where characteristics of the continuity equation are tangent to the boundary, which holds in particular for piecewise analytical boundaries.

Spectral instability of a steady flow of an incompressible viscous fluid past a rotating obstacle

Jiří Neustupa

Institute of Mathematics, Czech Academy of Sciences

Tuesday, 15. May 2018 - 9:00 to 10:00

We show that a steady solution U to the system of equations of motion of an incompressible Newtonian fluid past a rotating body is unstable if an associated linear operator L has at least one eigenvalue in the right half-plane in C. Our theorem does not directly follow from a series of preceding results on instability, mainly because the associated nonlinear operator is not bounded in the same space in which the instability is studied. As an important auxiliary result, we also show that the uniform growth bound of the C_0 semigroup e^{Lt} is equal to the spectral bound of operator L.