arXiv subjects

Vedran Sohinger

Publications and source records attributed to Vedran Sohinger.

23 records · Page 2Linked to original sources

arXiv2014

Randomization and the Gross-Pitaevskii hierarchy

We study the Gross-Pitaevskii hierarchy on the spatial domain $\mathbb{T}^3$. By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is $α>\frac{3}{4}$. It was shown in our previous joint work with Gressman that the range $α>1$ is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon, in which the spacetime estimate is known to hold whenever $α\geq 1$. The goal of our paper is to extend the range of $α$ in this class of estimates in a \emph{probabilistic sense}. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

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arXiv2014

Local existence of solutions to Randomized Gross-Pitaevskii hierarchies

In this paper, we study the local-in-time existence of solutions to randomized forms of the Gross-Pitaevskii hierarchy on periodic domains. In particular, we study the independently randomized Gross-Pitaevskii hierarchy and the dependently randomized Gross-Pitaevskii hierarchy, which were first introduced in the author's joint work with Staffilani \cite{SoSt}. For these hierarchies, we construct local-in-time low-regularity solutions in spaces which contain a random component. The constructed density matrices will solve the full randomized hierarchies, thus extending the results from \cite{SoSt}, where solutions solving arbitrarily long subhierarchies were given. Our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of T. Chen and Pavlović \cite{CP4}. The presence of randomization in the problem adds additional difficulties, most notably to estimating the Duhamel expansions that are crucial in the truncation argument. These difficulties are overcome by a detailed analysis of the Duhamel expansions. In the independently randomized case, we need to keep track of which randomization parameters appear in the Duhamel terms, whereas in the dependently randomized case, we express the Duhamel terms directly in terms of the initial data. In both cases, we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data.

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arXiv2014

Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.

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arXiv2014

A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbb{T}^3$ from the dynamics of many-body quantum systems

In this paper, we will obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on the three-dimensional torus $\mathbb{T}^3$ from the many-body limit of interacting bosonic systems. This type of result was previously obtained on $\mathbb{R}^3$ in the work of Erdős, Schlein, and Yau \cite{ESY2,ESY3,ESY4,ESY5}, and on $\mathbb{T}^2$ and $\mathbb{R}^2$ in the work of Kirkpatrick, Schlein, and Staffilani \cite{KSS}. Our proof relies on an unconditional uniqueness result for the Gross-Pitaevskii hierarchy at the level of regularity $α=1$, which is proved by using a modification of the techniques from the work of T. Chen, Hainzl, Pavlović and Seiringer \cite{ChHaPavSei} to the periodic setting. These techniques are based on the Quantum de Finetti theorem in the formulation of Ammari and Nier \cite{AmmariNier1,AmmariNier2} and Lewin, Nam, and Rougerie \cite{LewinNamRougerie}. In order to apply this approach in the periodic setting, we need to recall multilinear estimates obtained by Herr, Tataru, and Tzvetkov \cite{HTT}. Having proved the unconditional uniqueness result at the level of regularity $α=1$, we will apply it in order to finish the derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbb{T}^3$, which was started in the work of Elgart, Erdős, Schlein, and Yau \cite{EESY}. In the latter work, the authors obtain all the steps of Spohn's strategy for the derivation of the NLS \cite{Spohn}, except for the final step of uniqueness. Additional arguments are necessary to show that the objects constructed in \cite{EESY} satisfy the assumptions of the unconditional uniqueness theorem. Once we achieve this, we are able to prove the derivation result. In particular, we show \emph{Propagation of Chaos} for the defocusing Gross-Pitaevskii hierarchy on $\mathbb{T}^3$ for suitably chosen initial data.

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arXiv2014

The Boltzmann equation, Besov spaces, and optimal time decay rates in the whole space

We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\mathbb R}^{n}_x$ with $n \ge 3$, converge in large-time to the global Maxwellian with the optimal decay rate of $O(t^{-1/2(k+\varrho+\frac{n}{2}-\frac{n}{r})})$ in the $L^r_x(L^2_{v})$-norm for any $2\leq r\leq \infty$. These results hold for any $\varrho \in [0, n/2]$ as long as initially $\| f_0|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$. In the hard potential case, we prove faster decay results in the sense that if $|\mathbf{P} f_0\|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$ and $|({\mathbf{I} - \mathbf{P}}) f_0|_{\dot{B}^{-\varrho+1,\infty}_2 L^2_{v}} < \infty$ for $\varrho \in (n/2, (n+2)/2]$ then the solution decays to zero in $L^2_v(L^2_x)$ with the optimal large time decay rate of $O(t^{-1/2\varrho})$.

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