Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D
Daniela De Silva Nataša Pavlović Gigliola Staffilani Nikolaos Tzirakis
Discrete & Continuous Dynamical Systems - A 2007, 19(1): 37-65 doi: 10.3934/dcds.2007.19.37
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, $T^{d}_\lambda = R^{d}/{\lambda Z^{d}}$, $d=1,2$.
keywords: Global well-posedness nonlinear Schrödinger equation. nonlinear dispersive equations
Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions
Daniela De Silva Nataša Pavlović Gigliola Staffilani Nikolaos Tzirakis
Communications on Pure & Applied Analysis 2007, 6(4): 1023-1041 doi: 10.3934/cpaa.2007.6.1023
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.
keywords: global well-posedness Schrödinger equations $I$-method Morawetz estimates.
On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies
Thomas Chen Nataša Pavlović
Discrete & Continuous Dynamical Systems - A 2010, 27(2): 715-739 doi: 10.3934/dcds.2010.27.715
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.

keywords: Gross-Pitaevskii hierarchy Bose gas nonlinear Schrodinger equation.
Remarks concerning modified Navier-Stokes equations
Susan Friedlander Nataša Pavlović
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): 269-288 doi: 10.3934/dcds.2004.10.269
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.
keywords: dyadic model. Modified Navier-Stokes equations
An inviscid dyadic model of turbulence: The global attractor
Alexey Cheskidov Susan Friedlander Nataša Pavlović
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 781-794 doi: 10.3934/dcds.2010.26.781
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.
keywords: global attractor turbulence. Dyadic shell model

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