42 research outputs found
Radiating and non-radiating sources in elasticity
In this work, we study the inverse source problem of a fixed frequency for
the Navier's equation. We investigate that nonradiating external forces. If the
support of such a force has a convex or non-convex corner or edge on their
boundary, the force must be vanishing there. The vanishing property at corners
and edges holds also for sufficiently smooth transmission eigenfunctions in
elasticity. The idea originates from the enclosure method: The energy identity
and new type exponential solutions for the Navier's equation.Comment: 17 page
Inverse problem for wave equation with sources and observations on disjoint sets
We consider an inverse problem for a hyperbolic partial differential equation
on a compact Riemannian manifold. Assuming that and are
two disjoint open subsets of the boundary of the manifold we define the
restricted Dirichlet-to-Neumann operator . This
operator corresponds the boundary measurements when we have smooth sources
supported on and the fields produced by these sources are observed
on . We show that when and are disjoint but
their closures intersect at least at one point, then the restricted
Dirichlet-to-Neumann operator determines the
Riemannian manifold and the metric on it up to an isometry. In the Euclidian
space, the result yields that an anisotropic wave speed inside a compact body
is determined, up to a natural coordinate transformations, by measurements on
the boundary of the body even when wave sources are kept away from receivers.
Moreover, we show that if we have three arbitrary non-empty open subsets
, and of the boundary, then the restricted
Dirichlet-to-Neumann operators for determine the Riemannian manifold to an isometry. Similar result is proven
also for the finite-time boundary measurements when the hyperbolic equation
satisfies an exact controllability condition
Local exact controllability for the 2-D Boussinesq equations with the Navier slip boundary conditions
For distributed controls we get a local exact controllability for the 2-D Boussinesq equations in the case where the fluid is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions
Carleman estimate for the Navier-Stokes equations and applications
For linearized Navier-Stokes equations, we first derive a Carleman estimate with a regular weight function. Then we apply it to establish conditional stability for the
lateral Cauchy problem and finally we prove conditional stability estimates for the inverse source problem of determining a spatially varying divergence-free factor of a source term