84 research outputs found
A global existence result for a Keller-Segel type system with supercritical initial data
We consider a parabolic-elliptic Keller-Segel type system, which is related
to a simplified model of chemotaxis. Concerning the maximal range of existence
of solutions, there are essentially two kinds of results: either global
existence in time for general subcritical () initial data,
or blow--up in finite time for suitably chosen supercritical
() initial data with concentration around finitely many
points. As a matter of fact there are no results claiming the existence of
global solutions in the supercritical case. We solve this problem here and
prove that, for a particular set of initial data which share large
supercritical masses, the corresponding solution is global and uniformly
bounded
Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence
We are motivated by the study of the Microcanonical Variational Principle
within the Onsager's description of two-dimensional turbulence in the range of
energies where the equivalence of statistical ensembles fails. We obtain
sufficient conditions for the existence and multiplicity of solutions for the
corresponding Mean Field Equation on convex and "thin" enough domains in the
supercritical (with respect to the Moser-Trudinger inequality) regime. This is
a brand new achievement since existence results in the supercritical region
were previously known \un{only} on multiply connected domains. Then we study
the structure of these solutions by the analysis of their linearized problems
and also obtain a new uniqueness result for solutions of the Mean Field
Equation on thin domains whose energy is uniformly bounded from above. Finally
we evaluate the asymptotic expansion of those solutions with respect to the
thinning parameter and use it together with all the results obtained so far to
solve the Microcanonical Variational Principle in a small range of
supercritical energies where the entropy is eventually shown to be concave.Comment: 35 pages. In this version we have added an interesting remark (please
see Remark 1.17 p. 9). We have also slightly modified the statement of
Proposition 1.14 at p.8 so to include a part of it in a separate 4-line
Remark just after it (please see Remark 1.15 p.9
On the global bifurcation diagram of the Gel'fand problem
For domains of first kind [7,13] we describe the qualitative behavior of the
global bifurcation diagram of the unbounded branch of solutions of the Gel'fand
problem crossing the origin. At least to our knowledge this is the first result
about the exact monotonicity of the branch of non-minimal solutions which is
not just concerned with radial solutions [28] and/or with symmetric domains
[23]. Toward our goal we parametrize the branch not by the
-norm of the solutions but by the energy of the associated
mean field problem. The proof relies on a carefully modified spectral analysis
of mean field type equations.Comment: Intro has been expanded. References has been added. Minor expository
improvement
New universal estimates for free boundary problems arising in plasma physics
For a smooth and bounded domain, we derive a
sharp universal energy estimate for non-negative solutions of free boundary
problems on arising in plasma physics. As a consequence, we are able
to deduce new universal estimates for this class of problems. We first come up
with a sharp positivity threshold which guarantees that there is no free
boundary inside or either, equivalently, with a sharp necessary
condition for the existence of a free boundary in the interior of .
Then we derive an explicit bound for the -norm of non-negative
solutions and also obtain explicit estimates for the thresholds relative to
other neat density boundary values. At least to our knowledge, these are the
first explicit estimates of this sort in the superlinear case.Comment: 14 pages. arXiv admin note: text overlap with arXiv:2006.0477
Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions
We are concerned with the blow-up analysis of mean field equations. It has
been proven in [6] that solutions blowing-up at the same non-degenerate blow-up
set are unique. On the other hand, the authors in [18] show that solutions with
a degenerate blow-up set are in general non-unique. In this paper we first
prove that evenly symmetric solutions on a flat torus with a degenerate
two-point blow-up set are unique. In the second part of the paper we complete
the analysis by proving the existence of such blow-up solutions by using a
Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly
symmetric blow-up solutions come from one-point blow-up solutions of the mean
field equation on a "half" torus
On the global bifurcation diagram of the equation in dimension two
The aim of this note is to present the first qualitative global bifurcation
diagram of the equation . To this end, we
introduce the notion of domains of first/second kind for singular mean field
equations and base our approach on a suitable spectral analysis. In particular,
we treat also non-radial solutions and non-symmetric domains and show that the
shape of the branch of solutions still resembles the well-known one of the
model regular radial case on the disk. Some work is devoted also to the
asymptotic profile for .Comment: 15 page
A Courant nodal domain theorem for linearized mean field type equations
We are concerned with the analysis of a mean field type equation and its
linearization, which is a nonlocal operator, for which we estimate the number
of nodal domains for the radial eigenfunctions and the related uniqueness
properties.Comment: 18 page
Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains
The aim of this paper is to complete the program initiated in [50], [23] and
then carried out by several authors concerning non-degeneracy and uniqueness of
solutions to mean field equations. In particular, we consider mean field
equations with general singular data on non-smooth domains. The argument is
based on the Alexandrov-Bol inequality and on the eigenvalues analysis of
linearized singular Liouville-type problems
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