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### Open Access Journals

DCDS

The initial value problem for the $L^{2}$ critical semilinear
Schrödinger equation with periodic boundary data is considered.
We show that the problem is globally well-posed in $H^{s}(
T^{d} )$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively,
confirming in 2D a statement of Bourgain in [4].
We use the "$I$-method''. This method allows one to introduce a
modification of the energy functional that is well defined for
initial data below the $H^{1}(T^{d} )$ threshold. The main
ingredient in the proof is a "refinement" of the Strichartz's
estimates that hold true for solutions defined on the rescaled space,
$T^{d}_\lambda = R^{d}/{\lambda Z^{d}}$, $d=1,2$.

CPAA

The initial value problem for the $L^{2}$ critical semilinear
Schrödinger equation in $\mathbb R^n, n \geq 3$ is considered. We
show that the problem is globally well posed in $H^s(\mathbb
R^n )$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and when $1>s>
\frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$. We use the
"$I$-method" combined with a local in time Morawetz estimate.

DCDS

We consider the
dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$,
for cubic, quintic, focusing and defocusing interactions.
For both the focusing and defocusing case, and any $d\geq1$,
we prove local
existence and uniqueness of solutions in certain
Sobolev type spaces $\H_\xi^\alpha$ of sequences of marginal
density matrices which satisfy the space-time bound conjectured
by Klainerman and Machedon for the cubic GP hierarchy in $d=3$.
The regularity is accounted for by

$ \alpha $ > 1/2 if d=1

$ \alpha > \frac d2-\frac{1}{2(p-1)} if d\geq2 and (d,p)\neq(3,2) $

$ \alpha \geq 1 if (d,p)=(3,2) $

where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $\xi>0$ is arbitrary and determines the energy scale of the problem. For focusing GP hierarchies, we prove lower bounds on the blowup rate. Moreover, pseudoconformal invariance is established in the cases corresponding to $L^2$ criticality, both in the focusing and defocusing context. All of these results hold without the assumption of factorized initial conditions.

DCDS

We discuss some historical background concerning a modified version of the
Navier-Stokes equations for the motion of an incompressible fluid. The classical
(Newtonian) linear relation between the Cauchy stress tensor and the rate of
strain tensor yields the Navier-Stokes equations. Certain nonlinear relations are
also consistent with basic physical principals and result in equations with
"stronger" dissipation. We describe a class of models that has its genesis
in Kolmogorov's similarity hypothesis for 3-dimensional isotropic turbulence
and was formulated by Smagorinsky in the meteorological context of rapidly
rotating fluids and more generally by Ladyzhenskaya. These models also
describe the motion of fluids with shear dependent viscosities and have received
considerable attention. We present a dyadic model for such modified Navier-Stokes
equations. This model is an example of a hierarchical shell model.
Following the treatment of a (non-physically motivated) linear hyper-dissipative
model given by Katz-Pavlović, we prove for the dyadic model a bound for the
Hausdorff dimension of the singular set at the first time of blow up.
The result interpolates between the results of solvability for sufficiently
strong dissipation of Ladyzhenskaya, (later strengthened by
Nečas et al)
and the bound for the dimension of the singular set for the Navier-Stokes
equations proved by Caffarelli, Kohn and Nirenberg. We discuss the implications
of this dyadic model for the modified Navier-Stokes equation themselves.

DCDS

Properties of an infinite system of nonlinearly coupled
ordinary differential equations are discussed. This
system models some properties present in the equations
of motion for an inviscid fluid such as the skew symmetry
and the 3-dimensional scaling of the quadratic nonlinearity.
In a companion paper [8] it is proved that
every solution for the system with forcing
blows up in finite time in the Sobolev $H^{5/6}$ norm.
In this present paper, it is proved that after the
blow-up time all solutions stay in $H^s$, $s < 5/6$
for almost all time. It is proved that the model system
exhibits the phenomenon of anomalous (or turbulent) dissipation
which was conjectured for the Euler equations by Onsager.
As a consequence of this anomalous dissipation the unique equilibrium
of the system is a global attractor.

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