541 research outputs found
Dynamic Transitions for Quasilinear Systems and Cahn-Hilliard equation with Onsager mobility
The main objectives of this article are two-fold. First, we study the effect
of the nonlinear Onsager mobility on the phase transition and on the
well-posedness of the Cahn-Hilliard equation modeling a binary system. It is
shown in particular that the dynamic transition is essentially independent of
the nonlinearity of the Onsager mobility. However, the nonlinearity of the
mobility does cause substantial technical difficulty for the well-posedness and
for carrying out the dynamic transition analysis. For this reason, as a second
objective, we introduce a systematic approach to deal with phase transition
problems modeled by quasilinear partial differential equation, following the
ideas of the dynamic transition theory developed recently by Ma and Wang
Well-posedness of the Viscous Boussinesq System in Besov Spaces of Negative Order Near Index
This paper is concerned with well-posedness of the Boussinesq system. We
prove that the () dimensional Boussinesq system is well-psoed for
small initial data () either in
or in
if
, and , where
(, , )
is the logarithmically modified Besov space to the standard Besov space
. We also prove that this system is well-posed for small initial
data in
.Comment: 18 page
On the high-low method for NLS on the hyperbolic space
In this paper, we first prove that the cubic, defocusing nonlinear
Schr\"odinger equation on the two dimensional hyperbolic space with radial
initial data in is globally well-posed and scatters when . Then we extend the result to nonlineraities of order . The
result is proved by extending the high-low method of Bourgain in the hyperbolic
setting and by using a Morawetz type estimate proved by the first author and
Ionescu.Comment: The result is extended to general nonlineraitie
Polyharmonic approximation on the sphere
The purpose of this article is to provide new error estimates for a popular
type of SBF approximation on the sphere: approximating by linear combinations
of Green's functions of polyharmonic differential operators. We show that the
approximation order for this kind of approximation is for
functions having smoothness (for up to the order of the
underlying differential operator, just as in univariate spline theory). This is
an improvement over previous error estimates, which penalized the approximation
order when measuring error in , p>2 and held only in a restrictive setting
when measuring error in , p<2.Comment: 16 pages; revised version; to appear in Constr. Appro
Global Existence and Uniqueness of Solutions to the Maxwell-Schr{\"o}dinger Equations
The time local and global well-posedness for the Maxwell-Schr{\"o}dinger
equations is considered in Sobolev spaces in three spatial dimensions. The
Strichartz estimates of Koch and Tzvetkov type are used for obtaining the
solutions in the Sobolev spaces of low regularities. One of the main results is
that the solutions exist time globally for large data.Comment: 30 pages. In the revised version, the following modification was
made. (1) A line for dedication was added in the first page. (2) Some lines
were added at the bottom in page 4 and the top in page 5 in the first section
to make the description accurate. (3) Some typographical errors were
corrected throughout the pape
Fractional differentiability for solutions of nonlinear elliptic equations
We study nonlinear elliptic equations in divergence form
When
has linear growth in , and assuming that enjoys smoothness, local
well-posedness is found in for certain values of
and . In the particular case
, and ,
, we obtain for each
. Our main tool in the proof is a more general result, that
holds also if has growth in , , and
asserts local well-posedness in for each , provided that
satisfies a locally uniform condition
Local regularity for fractional heat equations
We prove the maximal local regularity of weak solutions to the parabolic
problem associated with the fractional Laplacian with homogeneous Dirichlet
boundary conditions on an arbitrary bounded open set
. Proofs combine classical abstract regularity
results for parabolic equations with some new local regularity results for the
associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756
Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces
We prove optimal integrability results for solutions of the p(x)-Laplace equation in the scale of (weak) Lebesgue spaces.
To obtain this, we show that variable exponent Riesz and Wolff potentials map L1 to variable exponent weak Lebesgue spaces
Stability of complex hyperbolic space under curvature-normalized Ricci flow
Using the maximal regularity theory for quasilinear parabolic systems, we
prove two stability results of complex hyperbolic space under the
curvature-normalized Ricci flow in complex dimensions two and higher. The first
result is on a closed manifold. The second result is on a complete noncompact
manifold. To prove both results, we fully analyze the structure of the
Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result,
we also define suitably weighted little H\"{o}lder spaces on a complete
noncompact manifold and establish their interpolation properties.Comment: Some typos in version 2 are correcte
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