356 research outputs found
Holder continuity for a drift-diffusion equation with pressure
We address the persistence of H\"older continuity for weak solutions of the
linear drift-diffusion equation with nonlocal pressure u_t + b \cdot \grad u
- \lap u = \grad p,\qquad \grad\cdot u =0 on ,
with . The drift velocity is assumed to be at the critical
regularity level, with respect to the natural scaling of the equations. The
proof draws on Campanato's characterization of H\"older spaces, and uses a
maximum-principle-type argument by which we control the growth in time of
certain local averages of . We provide an estimate that does not depend on
any local smallness condition on the vector field , but only on scale
invariant quantities
Local regularity for parabolic nonlocal operators
Weak solutions to parabolic integro-differential operators of order are studied. Local a priori estimates of H\"older norms and
a weak Harnack inequality are proved. These results are robust with respect to
. In this sense, the presentation is an extension of Moser's
result in 1971.Comment: 31 pages, 3 figure
Ricci flow of conformally compact metrics
In this paper we prove that given a smoothly conformally compact metric there
is a short-time solution to the Ricci flow that remains smoothly conformally
compact. We adapt recent results of Schn\"urer, Schulze and Simon to prove a
stability result for conformally compact Einstein metrics sufficiently close to
the hyperbolic metric.Comment: 26 pages, 2 figures. Version 2 includes stronger stability result and
fixes several typo
On a non-isothermal model for nematic liquid crystals
A model describing the evolution of a liquid crystal substance in the nematic
phase is investigated in terms of three basic state variables: the {\it
absolute temperature} \teta, the {\it velocity field} \ub, and the {\it
director field} \bd, representing preferred orientation of molecules in a
neighborhood of any point of a reference domain. The time evolution of the
velocity field is governed by the incompressible Navier-Stokes system, with a
non-isotropic stress tensor depending on the gradients of the velocity and of
the director field \bd, where the transport (viscosity) coefficients vary
with temperature. The dynamics of \bd is described by means of a parabolic
equation of Ginzburg-Landau type, with a suitable penalization term to relax
the constraint |\bd | = 1. The system is supplemented by a heat equation,
where the heat flux is given by a variant of Fourier's law, depending also on
the director field \bd. The proposed model is shown compatible with
\emph{First and Second laws} of thermodynamics, and the existence of
global-in-time weak solutions for the resulting PDE system is established,
without any essential restriction on the size of the data
Compactness and invariance properties of evolution operators associated with Kolmogorov operators with unbounded coefficients
In this paper we consider nonautonomous elliptic operators
with nontrivial potential term defined in , where is a
right-halfline (possibly ). We prove that we can associate an
evolution operator with in the space of all bounded
and continuous functions on . We also study the compactness
properties of the operator . Finally, we provide sufficient conditions
guaranteeing that each operator preserves the usual -spaces and
On the Amplitude Equations Arising at the Onset of the Oscillatory Instability in Pattern Formation
A well-known system of two amplitude equations is considered that describes the weakly nonlinear evolution of many nonequilibrium systems at the onset of the so-called oscillatory instability. Those equations depend on a small parameter, , that is a ratio between two distinguished spatial scales. In the limit , a simpler asymptotic model is obtained that consists of two complex cubic Ginzburg–Landau equations, coupled only by spatially averaged terms
A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation
We prove that every weak solution to the 3D Navier-Stokes equation that
belongs to the class and \n u belongs to localy
away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized
energy equality. In particular every such solution is suitable.Comment: 10 page
Optimization problem for extremals of the trace inequality in domains with holes
We study the Sobolev trace constant for functions defined in a bounded domain
\O that vanish in the subset We find a formula for the first variation
of the Sobolev trace with respect to hole. As a consequence of this formula, we
prove that when \O is a centered ball, the symmetric hole is critical when we
consider deformation that preserve volume but is not optimal for some case.Comment: 13 page
Gradient estimates for a degenerate parabolic equation with gradient absorption and applications
Qualitative properties of non-negative solutions to a quasilinear degenerate
parabolic equation with an absorption term depending solely on the gradient are
shown, providing information on the competition between the nonlinear diffusion
and the nonlinear absorption. In particular, the limit as time goes to infinity
of the mass of integrable solutions is identified, together with the rate of
expansion of the support for compactly supported initial data. The persistence
of dead cores is also shown. The proof of these results strongly relies on
gradient estimates which are first established
Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable
In this paper, we are concerned with the global wellposedness of 3-D
inhomogeneous incompressible Navier-Stokes equations \eqref{1.3} in the
critical Besov spaces with the norm of which are invariant by the scaling of
the equations and under a nonlinear smallness condition on the isentropic
critical Besov norm to the fluctuation of the initial density and the critical
anisotropic Besov norm of the horizontal components of the initial velocity
which have to be exponentially small compared with the critical anisotropic
Besov norm to the third component of the initial velocity. The novelty of this
results is that the isentropic space structure to the homogeneity of the
initial density function is consistent with the propagation of anisotropic
regularity for the velocity field. In the second part, we apply the same idea
to prove the global wellposedness of \eqref{1.3} with some large data which are
slowly varying in one direction
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