115 research outputs found
Analysis of Acoustic Wave Propagation in a Thin Moving Fluid
We study the propagation of acoustic waves in a fluid that is contained in a
thin two-dimensional tube, and that it is moving with a velocity profile that
only depends on the transversal coordinate of the tube. The governing equations
are the Galbrun equations, or, equivalently, the linearized Euler equations. We
analyze the approximate model that was recently derived by Bonnet-Bendhia,
Durufl\'e and Joly to describe the propagation of the acoustic waves in the
limit when the width of the tube goes to zero. We study this model for strictly
monotonic stable velocity profiles. We prove that the equations of the model of
Bonnet-Bendhia, Durufl\'e and Joly are well posed, i.e., that there is a unique
global solution, and that the solution depends continuously on the initial
data. Moreover, we prove that for smooth profiles the solution grows at most as
as , and that for piecewise linear profiles it grows at
most as . This establishes the stability of the model in a weak sense.
These results are obtained constructing a quasi-explicit representation of the
solution. Our quasi-explicit representation gives a physical interpretation of
the propagation of acoustic waves in the fluid and it provides an efficient way
to compute numerically the solution.Comment: 35 pages, 4 figure
Inverse Scattering at a Fixed Energy for Long-Range Potentials
In this paper we consider the inverse scattering problem at a fixed energy
for the Schr\"odinger equation with a long-range potential in \ere^d, d\geq
3. We prove that the long-range part can be uniquely reconstructed from the
leading forward singularity of the scattering amplitude at some positive
energy
High-Velocity Estimates for Schr\"odinger Operators in Two Dimensions: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering
We introduce a general class of long-range magnetic potentials and derive
high velocity limits for the scattering operators in quantum mechanics, in the
case of two dimensions. We analyze the high velocity limits in the presence of
an obstacle and we uniquely reconstruct from them the electric potential and
the magnetic field outside the obstacle. We also reconstruct the inaccessible
magnetic fluxes produced by fields inside the obstacle modulo . For
every magnetic potential in our class we prove that its behavior at
infinity () can be
characterized in a natural way. Under very general assumptions we prove that
can be uniquely
reconstructed for every . We characterize
properties of the support of the magnetic field outside the obstacle that
permit us to uniquely reconstruct either for all or for in a subset of
. We also give a wide class of magnetic fields outside the
obstacle allowing us to uniquely reconstruct the total magnetic flux (and
for all ). This
is relevant because, as it is well-known, in general the scattering operator
(even if is known for all velocities or energies) does not define uniquely the
total magnetic flux (and ). We analyze additionally
injectivity (i.e., uniqueness without giving a method for reconstruction) of
the high velocity limits of the scattering operator with respect to
. Assuming that the magnetic field outside the
obstacle is not identically zero, we provide a class of magnetic potentials for
which injectivity is valid
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