62 research outputs found
Compressible fluids interacting with a linear-elastic shell
We study the Navier--Stokes equations governing the motion of an isentropic
compressible fluid in three dimensions interacting with a flexible shell of
Koiter type. The latter one constitutes a moving part of the boundary of the
physical domain. Its deformation is modeled by a linearized version of Koiter's
elastic energy. We show the existence of weak solutions to the corresponding
system of PDEs provided the adiabatic exponent satisfies
( in two dimensions). The solution exists until the moving boundary
approaches a self-intersection. This provides a compressible counterpart of the
results in [D. Lengeler, M. \Ruzicka, Weak Solutions for an Incompressible
Newtonian Fluid Interacting with a Koiter Type Shell. Arch. Ration. Mech. Anal.
211 (2014), no. 1, 205--255] on incompressible Navier--Stokes equations
Self-improving property of the fast diffusion equation
We show that the gradient of the -power of a solution to a singular
parabolic equation of porous medium-type (also known as fast diffusion
equation), satisfies a reverse H\"older inequality in suitable intrinsic
cylinders. Relying on an intrinsic Calder\'on-Zygmund covering argument, we are
able to prove the local higher integrability of such a gradient for
. Our estimates are satisfied for a
general class of growth assumptions on the non linearity. In this way, we
extend the theory for (see [GS16] in the list of references) to the
singular case. In particular, an intrinsic metric that depends on the solution
itself is introduced for the singular regime.Comment: arXiv admin note: text overlap with arXiv:1603.0724
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