2,170 research outputs found
Critical Keller-Segel meets Burgers on : large-time smooth solutions
We show that solutions to the parabolic-elliptic Keller-Segel system on
with critical fractional diffusion
remain smooth for any initial data and any positive time. This disproves, at
least in the periodic setting, the large-data-blowup conjecture by Bournaveas
and Calvez. As a tool, we show smoothness of solutions to a modified critical
Burgers equation via a generalization of the method of moduli of continuity by
Kiselev, Nazarov and Shterenberg. over a setting where the considered equation
has no scaling. This auxiliary result may be interesting by itself. Finally, we
study the asymptotic behavior of global solutions, improving the existing
results.Comment: 17 page
Regularity and blow up for active scalars
We review some recent results for a class of fluid mechanics equations called
active scalars, with fractional dissipation. Our main examples are the surface
quasi-geostrophic equation, the Burgers equation, and the
Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods
which allow to prove existence of global regular solutions for the critical
dissipation. We also recall what is known about the possibility of finite time
blow up in the supercritical regime.Comment: 33 page
Global well-posedness for a slightly supercritical surface quasi-geostrophic equation
We use a nonlocal maximum principle to prove the global existence of smooth
solutions for a slightly supercritical surface quasi-geostrophic equation. By
this we mean that the velocity field is obtained from the active scalar
by a Fourier multiplier with symbol , where
is a smooth increasing function that grows slower than as
.Comment: 11 pages, second version with slightly stronger resul
Nonlocal maximum principles for active scalars
Active scalars appear in many problems of fluid dynamics. The most common
examples of active scalar equations are 2D Euler, Burgers, and 2D surface
quasi-geostrophic equations. Many questions about regularity and properties of
solutions of these equations remain open. We develop the idea of nonlocal
maximum principle, formulating a more general criterion and providing new
applications. The most interesting application is finite time regularization of
weak solutions in the supercritical regime.Comment: 19 page
A Bregman forward-backward linesearch algorithm for nonconvex composite optimization: superlinear convergence to nonisolated local minima
We introduce Bella, a locally superlinearly convergent Bregman forward
backward splitting method for minimizing the sum of two nonconvex functions,
one of which satisfying a relative smoothness condition and the other one
possibly nonsmooth. A key tool of our methodology is the Bregman
forward-backward envelope (BFBE), an exact and continuous penalty function with
favorable first- and second-order properties, and enjoying a nonlinear error
bound when the objective function satisfies a Lojasiewicz-type property. The
proposed algorithm is of linesearch type over the BFBE along candidate update
directions, and converges subsequentially to stationary points, globally under
a KL condition, and owing to the given nonlinear error bound can attain
superlinear convergence rates even when the limit point is a nonisolated
minimum, provided the directions are suitably selected
Stability of strings dual to flux tubes between static quarks in N=4 SYM
Computing heavy quark-antiquark potentials within the AdS/CFT correspondence
often leads to behaviors that differ from what one expects on general physical
grounds and field-theory considerations. To isolate the configurations of
physical interest, it is of utmost importance to examine the stability of the
string solutions dual to the flux tubes between the quark and antiquark. Here,
we formulate and prove several general statements concerning the perturbative
stability of such string solutions, relevant for static quark-antiquark pairs
in a general class of backgrounds, and we apply the results to N=4 SYM at
finite temperature and at generic points of the Coulomb branch. In all cases,
the problematic regions are found to be unstable and hence physically
irrelevant.Comment: 43 pages, 18 figures; v3: typos correcte
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