22 research outputs found
On the interaction between quasilinear elastodynamics and the Navier-Stokes equations
The interaction between a viscous fluid and an elastic solid is modeled by a
system of parabolic and hyperbolic equations, coupled to one another along the
moving material interface through the continuity of the velocity and traction
vectors. We prove the existence and uniqueness (locally in time) of strong
solutions in Sobolev spaces for quasilinear elastodynamics coupled to the
incompressible Navier-Stokes equations along a moving interface. Unlike our
approach for the case of linear elastodynamics, we cannot employ a fixed-point
argument on the nonlinear system itself, and are instead forced to regularize
it by a particular parabolic artificial viscosity term. We proceed to show that
with this specific regularization, we obtain a time interval of existence which
is independent of the artificial viscosity; together with a priori estimates,
we identify the global solution (in both phases), as well as the interface
motion, as a weak limit in srong norms of our sequence of regularized problems.Comment: 43 pages, to appear in Archive for Rational Mechanics and Analysi
Some Results on the Boundary Control of Systems of Conservation Laws
This note is concerned with the study of the initial boundary value problem
for systems of conservation laws from the point of view of control theory,
where the initial data is fixed and the boundary data are regarded as control
functions. We first consider the problem of controllability at a fixed time for
genuinely nonlinear Temple class systems, and present a description of the set
of attainable configurations of the corresponding solutions in terms of
suitable Oleinik-type estimates. We next present a result concerning the
asymptotic stabilization near a constant state for general systems.
Finally we show with an example that in general one cannot achieve exact
controllability to a constant state in finite time.Comment: 10 pages, 4 figures, conferenc
The boundary Riemann solver coming from the real vanishing viscosity approximation
We study a family of initial boundary value problems associated to mixed
hyperbolic-parabolic systems:
v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x =
\epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx}
The conservative case is, in particular, included in the previous
formulation.
We suppose that the solutions to these problems converge to a
unique limit. Also, it is assumed smallness of the total variation and other
technical hypotheses and it is provided a complete characterization of the
limit.
The most interesting points are the following two.
First, the boundary characteristic case is considered, i.e. one eigenvalue of
can be .
Second, we take into account the possibility that is not invertible. To
deal with this case, we take as hypotheses conditions that were introduced by
Kawashima and Shizuta relying on physically meaningful examples. We also
introduce a new condition of block linear degeneracy. We prove that, if it is
not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added
Section 3.1.2. Minor changes in other section
Degenerate nonlinear hyperbolic conservation laws: non classical geometrical theory
International audienceKinetic selection principles have been shown to be useful and physically reasonable in nonlinear hyperbolic problems with large amplitude phase transitions. We refer to Abeyaratne and Knowles, [A-K], for discussion on the subject. They also have been used for degenerate nonlinear problems, where the genuine nonlinearity property of Lax is violated. This is the framework of Liu, [Li], in the kinetic situation of Hayes and Le Floch, [H-L], where their so-called non classical shocks may be seen as small amplitude phase transitions. Here, we describe the local geometry generated by the generic non genuinely nonlinear assumption. A geometric kinetic criterion can be used to select indeterminate simple waves and obtain the well-posedness of the local Riemann problem. A particular case is the entropic kinetic criterion of Hayes and Le Floch