171 research outputs found
Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions
In connection with the weak null condition, Alinhac introduced a sufficient
condition for global existence of small amplitude solutions to systems of
semilinear wave equations in three space dimensions. We introduce a slightly
weaker sufficient condition for the small data global existence, and we
investigate the asymptotic pointwise behavior of global solutions for systems
satisfying this condition. As an application, the asymptotic behavior of global
solutions under the Alinhac condition is also derived.Comment: 56 pages, the final versio
Global Solutions for Incompressible Viscoelastic Fluids
We prove the existence of both local and global smooth solutions to the
Cauchy problem in the whole space and the periodic problem in the n-dimensional
torus for the incompressible viscoelastic system of Oldroyd-B type in the case
of near equilibrium initial data. The results hold in both two and three
dimensional spaces. The results and methods presented in this paper are also
valid for a wide range of elastic complex fluids, such as magnetohydrodynamics,
liquid crystals and mixture problems.Comment: We prove the existence of global smooth solutions to the Cauchy
problem for the incompressible viscoelastic system of Oldroyd-B type in the
case of near equilibrium initial dat
Generalized harmonic spatial coordinates and hyperbolic shift conditions
We propose a generalization of the condition for harmonic spatial coordinates
analogous to the generalization of the harmonic time slices introduced by Bona
et al., and closely related to dynamic shift conditions recently proposed by
Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic
spatial coordinates imply a condition for the shift vector that has the form of
an evolution equation for the shift components. We find that in order to
decouple the slicing condition from the evolution equation for the shift it is
necessary to use a rescaled shift vector. The initial form of the generalized
harmonic shift condition is not spatially covariant, but we propose a simple
way to make it fully covariant so that it can be used in coordinate systems
other than Cartesian. We also analyze the effect of the shift condition
proposed here on the hyperbolicity of the evolution equations of general
relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible
development of blow-ups. Finally, we perform a series of numerical experiments
to illustrate the behavior of this shift condition.Comment: 18 pages and 12 figures, extensively revised version explaining in
the new Section IV how the shift condition can be made 3-covarian
Wave equation with concentrated nonlinearities
In this paper we address the problem of wave dynamics in presence of
concentrated nonlinearities. Given a vector field on an open subset of
\CO^n and a discrete set Y\subset\RE^3 with elements, we define a
nonlinear operator on L^2(\RE^3) which coincides with the free
Laplacian when restricted to regular functions vanishing at , and which
reduces to the usual Laplacian with point interactions placed at when
is linear and is represented by an Hermitean matrix. We then consider the
nonlinear wave equation and study the
corresponding Cauchy problem, giving an existence and uniqueness result in the
case is Lipschitz. The solution of such a problem is explicitly expressed
in terms of the solutions of two Cauchy problem: one relative to a free wave
equation and the other relative to an inhomogeneous ordinary differential
equation with delay and principal part . Main properties of
the solution are given and, when is a singleton, the mechanism and details
of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and
General, special issue on Singular Interactions in Quantum Mechanics:
Solvable Model
A simple method for finite range decomposition of quadratic forms and Gaussian fields
We present a simple method to decompose the Green forms corresponding to a
large class of interesting symmetric Dirichlet forms into integrals over
symmetric positive semi-definite and finite range (properly supported) forms
that are smoother than the original Green form. This result gives rise to
multiscale decompositions of the associated Gaussian free fields into sums of
independent smoother Gaussian fields with spatially localized correlations. Our
method makes use of the finite propagation speed of the wave equation and
Chebyshev polynomials. It improves several existing results and also gives
simpler proofs.Comment: minor correction for t<
Exponential decay for the damped wave equation in unbounded domains
We study the decay of the semigroup generated by the damped wave equation in
an unbounded domain. We first prove under the natural geometric control
condition the exponential decay of the semigroup. Then we prove under a weaker
condition the logarithmic decay of the solutions (assuming that the initial
data are smoother). As corollaries, we obtain several extensions of previous
results of stabilisation and control
Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers
Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the
shock wave case, we study stability of compressive, or "shock-like", boundary
layers of the isentropic compressible Navier-Stokes equations with gamma-law
pressure by a combination of asymptotic ODE estimates and numerical Evans
function computations. Our results indicate stability for gamma in the interval
[1, 3] for all compressive boundary-layers, independent of amplitude, save for
inflow layers in the characteristic limit (not treated). Expansive inflow
boundary-layers have been shown to be stable for all amplitudes by Matsumura
and Nishihara using energy estimates. Besides the parameter of amplitude
appearing in the shock case, the boundary-layer case features an additional
parameter measuring displacement of the background profile, which greatly
complicates the resulting case structure. Moreover, inflow boundary layers turn
out to have quite delicate stability in both large-displacement and
large-amplitude limits, necessitating the additional use of a mod-two stability
index studied earlier by Serre and Zumbrun in order to decide stability
Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the "Double Null Foliation Gauge"
The main goal of this work consists in showing that the analytic solutions
for a class of characteristic problems for the Einstein vacuum equations have
an existence region larger than the one provided by the Cauchy-Kowalevski
theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove
this result we first describe a geometric way of writing the vacuum Einstein
equations for the characteristic problems we are considering, in a gauge
characterized by the introduction of a double null cone foliation of the
spacetime. Then we prove that the existence region for the analytic solutions
can be extended to a larger region which depends only on the validity of the
apriori estimates for the Weyl equations, associated to the "Bel-Robinson
norms". In particular if the initial data are sufficiently small we show that
the analytic solution is global. Before showing how to extend the existence
region we describe the same result in the case of the Burger equation, which,
even if much simpler, nevertheless requires analogous logical steps required
for the general proof. Due to length of this work, in this paper we mainly
concentrate on the definition of the gauge we use and on writing in a
"geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski
theorem is adapted to the characteristic problem for the Einstein equations and
we describe how the existence region can be extended in the case of the Burger
equation. Finally we describe the structure of the extension proof in the case
of the Einstein equations. The technical parts of this last result is the
content of a second paper.Comment: 68 page
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