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Nataša Pavlović

Publications and source records attributed to Nataša Pavlović.

44 records · Page 3Linked to original sources
arXiv2009

On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies

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 $\cH_ξ^α$ of sequences of marginal density matrices. The regularity is accounted for by $α>\frac12& if $d=1$, $α>\frac d2-\frac{1}{2(p-1)} $ if $d\geq2$ and $(d,p)\neq(3,2)$, and $α\geq1$ if $(d,p)=(3,2)$, where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $ξ>0$ is arbitrary and determines the energy scale of the problem. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in $d=3$. In the defocusing case, we prove the existence and uniqueness of solutions globally in time for the cubic GP hierarchy for $1\leq d\leq3$, and of the quintic GP hierarchy for $1\leq d\leq 2$, in an appropriate space of Sobolev type, and under the assumption of an a priori energy bound. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. Also 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.

math-ph↗
arXiv2008

Ill-posedness of the Navier-Stokes equations in a critical space in 3D

We prove that the Cauchy problem for the three dimensional Navier-Stokes equations is ill posed in $\dot{B}^{-1,\infty}_{\infty}$ in the sense that a ``norm inflation'' happens in finite time. More precisely, we show that initial data in the Schwartz class $\mathcal{S}$ that are arbitrarily small in $\dot{B}^{-1, \infty}_{\infty}$ can produce solutions arbitrarily large in $\dot{B}^{-1, \infty}_{\infty}$ after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in $\dot{B}^{-1, \infty}_{\infty}$ at the origin.

math.AP↗
arXiv2007

Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space $H^{s}(\mathbb R)$ for any $s>{1/3}$. This improves the result in \cite{tz}, where global well-posedness was established for any $s>{4/9}$. We use the $I$-method to take advantage of the conservation laws of the equation. The new ingredient in our proof is an interaction Morawetz estimate for the smoothed out solution $Iu$. As a byproduct of our proof we also obtain that the $H^{s}$ norm of the solution obeys polynomial-in-time bounds.

math.AP↗
arXiv2007

Regularity of solutions to the Navier-Stokes equations evolving from small data in BMO^{-1}

In 2001, H. Koch and D. Tataru proved the existence of global in time solutions to the incompressible Navier-Stokes equations in ${\mathbb{R}}^d$ for initial data small enough in $BMO^{-1}$. We show in this article that the Koch and Tataru solution has higher regularity. As a consequence, we get a decay estimate in time for any space derivative, and space analyticity of the solution. Also as an application of our regularity theorem, we prove a regularity result for self-similar solutions.

math.AP↗
arXiv2006

Global Well-Posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

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 $H^{s}({\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \cite{bo2}. 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 $H^{1}({\Bbb T^{d}})$ 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, $\Bbb T^{d}_λ = \Bbb R^{d}/{λ\Bbb Z^{d}}$, $d=1,2$.

math.AP↗
arXiv2005

Nonlinear instability for the Navier-Stokes equations

It is proved, using a bootstrap argument, that linear instability implies nonlinear instability for the incompressible Navier-Stokes equations in $L^p$ for all $p \in (1,\infty)$ and any finite or infinite domain in any dimension $n$.

math.AP↗