312 research outputs found
Vanishing viscosity limit for an expanding domain in space
We study the limiting behavior of viscous incompressible flows when the fluid
domain is allowed to expand as the viscosity vanishes. We describe precise
conditions under which the limiting flow satisfies the full space Euler
equations. The argument is based on truncation and on energy estimates,
following the structure of the proof of Kato's criterion for the vanishing
viscosity limit. This work complements previous work by the authors, see
[Kelliher, Comm. Math. Phys. 278 (2008), 753-773] and [arXiv:0801.4935v1].Comment: 23 pages, submitted for publicatio
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
Decay and Continuity of Boltzmann Equation in Bounded Domains
Boundaries occur naturally in kinetic equations and boundary effects are
crucial for dynamics of dilute gases governed by the Boltzmann equation. We
develop a mathematical theory to study the time decay and continuity of
Boltzmann solutions for four basic types of boundary conditions: inflow,
bounce-back reflection, specular reflection, and diffuse reflection. We
establish exponential decay in norm for hard potentials for
general classes of smooth domains near an absolute Maxwellian. Moreover, in
convex domains, we also establish continuity for these Boltzmann solutions away
from the grazing set of the velocity at the boundary. Our contribution is based
on a new decay theory and its interplay with delicate
decay analysis for the linearized Boltzmann equation, in the presence of many
repeated interactions with the boundary.Comment: 89 pages
Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics
For the equations of elastodynamics with polyconvex stored energy, and some
related simpler systems, we define a notion of dissipative measure-valued
solution and show that such a solution agrees with a classical solution with
the same initial data when such a classical solution exists. As an application
of the method we give a short proof of strong convergence in the continuum
limit of a lattice approximation of one dimensional elastodynamics in the
presence of a classical solution. Also, for a system of conservation laws
endowed with a positive and convex entropy, we show that dissipative
measure-valued solutions attain their initial data in a strong sense after time
averaging
Stable ground states for the relativistic gravitational Vlasov-Poisson system
We consider the three dimensional gravitational Vlasov-Poisson (GVP) system
in both classical and relativistic cases. The classical problem is subcritical
in the natural energy space and the stability of a large class of ground states
has been derived by various authors. The relativistic problem is critical and
displays finite time blow up solutions. Using standard concentration
compactness techniques, we however show that the breaking of the scaling
symmetry allows the existence of stable relativistic ground states. A new
feature in our analysis which applies both to the classical and relativistic
problem is that the orbital stability of the ground states does not rely as
usual on an argument of uniqueness of suitable minimizers --which is mostly
unknown-- but on strong rigidity properties of the transport flow, and this
extends the class of minimizers for which orbital stability is now proved
Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations
In this article we study the fractal Navier-Stokes equations by using
stochastic Lagrangian particle path approach in Constantin and Iyer
\cite{Co-Iy}. More precisely, a stochastic representation for the fractal
Navier-Stokes equations is given in terms of stochastic differential equations
driven by L\'evy processes. Basing on this representation, a self-contained
proof for the existence of local unique solution for the fractal Navier-Stokes
equation with initial data in \mW^{1,p} is provided, and in the case of two
dimensions or large viscosity, the existence of global solution is also
obtained. In order to obtain the global existence in any dimensions for large
viscosity, the gradient estimates for L\'evy processes with time dependent and
discontinuous drifts is proved.Comment: 19 page
Nonlinear hyperbolic systems: Non-degenerate flux, inner speed variation, and graph solutions
We study the Cauchy problem for general, nonlinear, strictly hyperbolic
systems of partial differential equations in one space variable. First, we
re-visit the construction of the solution to the Riemann problem and introduce
the notion of a nondegenerate (ND) system. This is the optimal condition
guaranteeing, as we show it, that the Riemann problem can be solved with
finitely many waves, only; we establish that the ND condition is generic in the
sense of Baire (for the Whitney topology), so that any system can be approached
by a ND system. Second, we introduce the concept of inner speed variation and
we derive new interaction estimates on wave speeds. Third, we design a wave
front tracking scheme and establish its strong convergence to the entropy
solution of the Cauchy problem; this provides a new existence proof as well as
an approximation algorithm. As an application, we investigate the
time-regularity of the graph solutions introduced by the second author,
and propose a geometric version of our scheme; in turn, the spatial component
of a graph solution can be chosen to be continuous in both time and space,
while its component is continuous in space and has bounded variation in
time.Comment: 74 page
Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials
In this paper it is shown that unique solutions to the relativistic Boltzmann
equation exist for all time and decay with any polynomial rate towards their
steady state relativistic Maxwellian provided that the initial data starts out
sufficiently close in . If the initial data are continuous then
so is the corresponding solution. We work in the case of a spatially periodic
box. Conditions on the collision kernel are generic in the sense of
(Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves
the open question of global existence for the soft potentials.Comment: 64 page
On the selection of the classical limit for potentials with BV derivatives
We consider the classical limit of the quantum evolution, with some rough
potential, of wave packets concentrated near singular trajectories of the
underlying dynamics. We prove that under appropriate conditions, even in the
case of BV vector fields, the correct classical limit can be selected
A Blow-Up Criterion for Classical Solutions to the Compressible Navier-Stokes Equations
In this paper, we obtain a blow up criterion for classical solutions to the
3-D compressible Naiver-Stokes equations just in terms of the gradient of the
velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible
flow. In addition, initial vacuum is allowed in our case.Comment: 25 page
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