Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Formation of condensations for non-radial solutions to 3-wave kinetic equations

Gigliola Staffilani, Minh-Binh Tran

Abstract

We consider in this work a $2$-dimensional $3$-wave kinetic equation describing the dynamics of the thermal cloud outside a Bose-Einstein Condensate. We construct global non-radial mild solutions for the equation. Those mild solutions are the summation of Dirac masses on circles. We prove that in each spatial direction, either Dirac masses at the origin, which are the so-called Bose-Einstein condensates, can be formed in finite time or the solutions converge to Bose-Einstein condensates as time evolves to infinity. We also describe a dynamics of the formation of the Bose-Einstein condensates latter case. In this case, on each direction, the solutions accumulate around circles close to the origin at growth rates at least linearly in time.

Formation of condensations for non-radial solutions to 3-wave kinetic equations

Abstract

We consider in this work a -dimensional -wave kinetic equation describing the dynamics of the thermal cloud outside a Bose-Einstein Condensate. We construct global non-radial mild solutions for the equation. Those mild solutions are the summation of Dirac masses on circles. We prove that in each spatial direction, either Dirac masses at the origin, which are the so-called Bose-Einstein condensates, can be formed in finite time or the solutions converge to Bose-Einstein condensates as time evolves to infinity. We also describe a dynamics of the formation of the Bose-Einstein condensates latter case. In this case, on each direction, the solutions accumulate around circles close to the origin at growth rates at least linearly in time.

Paper Structure

This paper contains 11 sections, 10 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The 3-wave kinetic equation
  3. Physical context of the model \ref{['3wave']}-\ref{['E1']}-\ref{['E3']}-\ref{['E2']}
  4. The Settings and main results
  5. Measure space
  6. Main theorem
  7. Estimates for the collision operator and a local existence result
  8. Moment estimates
  9. Long time limit
  10. Proof of Theorem \ref{['maintheorem']}
  11. Appendix

Key Result

Lemma 1

For any positive constant $R>0$, we define set Suppose that $\{f_n\}_{n=0}^\infty$ is a sequence in $A$ that converges to $f$ in the weak topology of $\mathcal{M}_+([0,\infty))$ as $n$ goes to infinity, for a.e. $\widehat{k}\in \mathbb{S}$, then $f\in A$.

Figures (1)

  • Figure 1: The circular lattice $\Lambda$. The initial conditions are supported on the red circles.

Theorems & Definitions (21)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Theorem 2
  • Remark 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • ...and 11 more