64 research outputs found
On a non-solenoidal approximation to the incompressible Navier-Stokes equations
We establish an asymptotic profile that sharply describes the behavior as
for solutions to a non-solenoidal approximation of the
incompressible Navier-Stokes equations introduced by Temam. The solutions of
Temam's model are known to converge to the corresponding solutions of the
classical Navier-Stokes, e.g., in , provided
, where is the physical parameter related to the
artificial compressibility term. However, we show that such model is no longer
a good approximation of Navier-Stokes for large times: indeed, its solutions
can decay much slower as than the corresponding solutions of
Navier-Stokes.Comment: Submitted to the Journal of the London Mathematical Society (under
revision
Space-time decay of Navier-Stokes flows invariant under rotations
We show that the solutions to the non-stationary Navier-Stokes equations in
, which are left invariant under the action of discrete subgroups
of the orthogonal group decay much faster as or than in the generic case and we compute, for each subgroup, the
precise decay rates in space-time of the velocity field
A Liouville theorem for the Degasperis-Procesi equation
We prove that the only global, strong, spatially periodic solution to the
Degasperis-Procesi equation, vanishing at some point (t0, x0), is the
identically zero solution. We also establish the analogue of such
Liouville-type theorem for the Degasperis-Procesi equation with an additional
dispersive term.Comment: Post-refereeing version. To appear on Annali Scienze Scuola Norm.
Sup. Pisa. Doi 10.2422/2036-2145.201410\_01
Concentration-diffusion effects in viscous incompressible flows
Given a finite sequence of times , we construct an example of
a smooth solution of the free nonstationnary Navier--Stokes equations in
, , such that: (i) The velocity field is spatially poorly
localized at the beginning of the evolution but tends to concentrate until, as
the time approaches , it becomes well-localized. (ii) Then spreads
out again after , and such concentration-diffusion phenomena are later
reproduced near the instants , , ...Comment: Indiana Univ. Math. Journal (to appear
Global existence versus blow up for some models of interacting particles
We study the global existence and space-time asymptotics of solutions for a
class of nonlocal parabolic semilinear equations. Our models include the
Nernst-Planck and the Debye-Hukel drift-diffusion systems as well as
parabolic-elliptic systems of chemotaxis. In the case of a model of
self-gravitating particles, we also give a result on the finite time blow up of
solutions with localized and oscillating complex-valued initial data, using a
method by S. Montgomery-Smith.Comment: Colloq. Math. (to appear
On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics
We study the behavior at infinity, with respect to the space variable, of
solutions to the magnetohydrodynamics equations in . We prove that
if the initial magnetic field decays sufficiently fast, then the plasma flow
behaves as a solution of the free nonstationnary Navier--Stokes equations when
, and that the magnetic field will govern the decay of the
plasma, if it is poorly localized at the beginning of the evolution. Our main
tools are boundedness criteria for convolution operators in weighted spaces.Comment: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics
(to appear) (0000) --xx-
New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System
We show that solutions of the non-stationnary incompressible
Navier--Stokes system in () starting from mild decaying data
behave as as a potential field: u(x,t) = e^{t\Delta}a(x) +
\gamma_d\nabla_x(\sum_{h,k} \frac{\delta_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}}
K_{h,k}(t))+\mathfrak{o}(\frac{1}{|x|^{d+1}}) where is a constant
and is the energy matrix of the flow. We
deduce that, for well localized data, and for small and large enough ,
c t |x|^{-(d+1)} \le |u(x,t)|\le c' t |x|^{-(d+1)}, where the lower bound holds
on the complementary of a set of directions, of arbitrary small measure on
. We also obtain new lower bounds for the large time decay of
the weighted- norms, extending previous results of Schonbek, Miyakawa, Bae
and Jin.Comment: 26 pages, article to appear in Journal de Math\'ematiques Pures et
Appliqu\'ee
On permanent and breaking waves in hyperelastic rods and rings
We prove that the only global strong solution of the periodic rod equation
vanishing in at least one point is the identically zero solution.
Such conclusion holds provided the physical parameter of the model
(related to the finger deformation tensor) is outside some neighborhood of the
origin and applies in particular for the Camassa--Holm equation, corresponding
to . We also establish the analogue of this unique continuation
result in the case of non-periodic solutions defined on the whole real line
with vanishing boundary conditions at infinity. Our analysis relies on the
application of new local-in-space blowup criteria and involves the computation
of several best constants in convolution estimates and weighted Poincar\'e
inequalities.Comment: Corrected proofs. To appear on J. Funct. Ana
Blowup issues for a class of nonlinear dispersive wave equations
In this paper we consider the nonlinear dispersive wave equation on the real
line,
,
that for appropriate choices of the functions and includes well known
models, such as Dai's equation for the study of vibrations inside elastic rods
or the Camassa--Holm equation modelling water wave propagation in shallow
water. We establish a local-in-space blowup criterion (i.e., a criterion
involving only the properties of the data in a neighbourhood of a single
point) simplifying and extending earlier blowup criteria for this equation. Our
arguments apply both to the finite and infinite energy case, yielding the
finite time blowup of strong solutions with possibly different behavior as
and
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