57 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
Ladyzhenskaya-Prodi-Serrin condition for fluid-structure interaction systems
We consider the interaction of a viscous incompressible fluid with a flexible
shell in three space dimensions. The fluid is described by the
three-dimensional incompressible Navier--Stokes equations in a domain that is
changing in accordance with the motion of the structure. The displacement of
the latter evolves along a visco-elastic shell equation. Both are coupled
through kinematic boundary conditions and the balance of forces.
We prove a counterpart of the classical Ladyzhenskaya-Prodi-Serrin condition
yielding conditional regularity and uniqueness of a solution.
Our result is a consequence of the following three ingredients which might be
of independent interest: {\bf (i)} the existence of local strong solutions,
{\bf (ii)} an acceleration estimate (under the Serrin assumption) ultimately
controlling the second-order energy norm, and {\bf (iii)} a weak-strong
uniqueness theorem. The first point, and to some extent, the last point were
previously known for the case of elastic plates, which means that the relaxed
state is flat. We extend these results to the case of visco-elastic shells,
which means that more general reference geometries are considered such as
cylinders or spheres. The second point, i.e. the acceleration estimate for
three-dimensional fluids is new even in the case of plates.Comment: 42 page
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