1,680 research outputs found
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher
This review is dedicated to recent results on the 2d parabolic-elliptic
Patlak-Keller-Segel model, and on its variant in higher dimensions where the
diffusion is of critical porous medium type. Both of these models have a
critical mass such that the solutions exist globally in time if the mass
is less than and above which there are solutions which blowup in finite
time. The main tools, in particular the free energy, and the idea of the
methods are set out. A number of open questions are also stated.Comment: 26 page
Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential
We consider a cubic nonlinear Schroedinger equation with periodic potential.
In a semiclassical scaling the nonlinear interaction of modulated pulses
concentrated in one or several Bloch bands is studied. The notion of closed
mode systems is introduced which allows for the rigorous derivation of a finite
system of amplitude equations describing the macroscopic dynamics of these
pulses
Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation
inequalities (CKN), with two radial power law weights and exponents in a
subcritical range. We address the question of symmetry breaking: are the
optimal functions radially symmetric, or not ? Our intuition comes from a
weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy
- entropy production inequality which governs the intermediate asymptotics is
indeed equivalent to (CKN), and the self-similar profiles are optimal for
(CKN). We establish an explicit symmetry breaking condition by proving the
linear instability of the radial optimal functions for (CKN). Symmetry breaking
in (CKN) also has consequences on entropy - entropy production inequalities and
on the intermediate asymptotics for (WFD). Even when no symmetry holds in
(CKN), asymptotic rates of convergence of the solutions to (WFD) are determined
by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a
linearized entropy - entropy production inequality. All our results rely on the
study of the bottom of the spectrum of the linearized diffusion operator around
the self-similar profiles, which is equivalent to the linearization of (CKN)
around the radial optimal functions, and on variational methods. Consequences
for the (WFD) flow will be studied in Part II of this work
Global Well-Posedness Of A Non-Local Burgers Equation: The Periodic Case
This paper is concerned with the study of a non-local Burgers equation for
positive bounded periodic initial data. The equation reads We construct global classical solutions starting from
smooth positive data, and global weak solutions starting from data in
. We show that any weak solution is instantaneously regularized into
. We also describe the long-time behavior of all solutions. Our
methods follow several recent advances in the regularity theory of parabolic
integro-differential equations.Comment: 27 pages, 11 figure
Conditional regularity of solutions of the three dimensional Navier-Stokes equations and implications for intermittency
Two unusual time-integral conditional regularity results are presented for
the three-dimensional Navier-Stokes equations. The ideas are based on
-norms of the vorticity, denoted by , and particularly
on , where for . The first result, more appropriate for the unforced case, can be stated
simply : if there exists an for which the integral condition
is satisfied () then no singularity can occur on . The
constant for large . Secondly, for the forced case, by
imposing a critical \textit{lower} bound on , no
singularity can occur in for \textit{large} initial data. Movement
across this critical lower bound shows how solutions can behave intermittently,
in analogy with a relaxation oscillator. Potential singularities that drive
over this critical value can be ruled out whereas
other types cannot.Comment: A frequency was missing in the definition of D_{m} in (I5) v3. 11
pages, 1 figur
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