Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

1 Some elementary examples


2 Optimal transport plans: existence and regularity


3 The one dimensional case


4 The ODE version of the optimal transport problem


5 The PDE version of the optimal transport problem and the p-laplacian approximation


6 Existence of optimal transport maps


7 Regularity and uniqueness of the transport density


8 The Bouchitte-Buttazzo mass optimization problem


9 Appendix: some measure theoretic results


References

Lecture Notes on Optimal Transport Problems

In this paper, we mainly review recent results on mathematical theory and
numerical methods for Bose-Einstein condensation (BEC),
based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then
carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for
spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

http://osada.csp.escience.cn/dct/attach/Y2xiOmNsYjpwZGY6NjI5NzY=

Mathematical theory and numerical methods for Bose-Einstein condensation

We present a method to prove convergence of gradient flows of families of energies that Γ-converge to a limiting energy. It provides lower-bound criteria to obtain the convergence that correspond to a sort of C1-order Γ-convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field. © 2004 Wiley Periodicals, Inc.

Gamma-convergence of gradient flows with applications to Ginzburg-Landau

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

We study the Ginzburg-Landau functional $$I_{\epsilon}(u) := \frac{1}{\ln(1/\epsilon)}\int_U \frac12 |\nabla u|^2 + \frac{1}{4 \epsilon^2} (1-|u|^2)^2 dx ,$$
 for $u\in H^1(U; \R^2)$
 , where U is a bounded, open subset of $\mathbb{R}{2}$
 . We show that if a sequence of functions $u^\epsilon$
 satisfies $\sup I_{\epsilon}(u^\epsilon)< \infty$
 , then their Jacobians $Ju^\epsilon$
 are precompact in the dual of $C^{0,\alpha}_c$
 for every $\alpha \in (0,1]$
 . Moreover, any limiting measure is a sum of point masses. We also characterize the $\Gamma$
 -limit $I(\cdot)$
 of the functionals $I_{\epsilon}(\cdot)$
 , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if $u \in B2V(U;S^1)$
 , and for $ u \in B2V(U;S^1), I(u)$
 is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if $I_\epsilon(u^\epsilon) \le C$
 then the Jacobians $Ju^\epsilon$
 are again precompact in $\left(C^{0,\alpha}_c\right)^*$
 for all $\alpha\in (0,1]$
 , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the $\Gamma$
 limit of the Ginzburg-Landau functional.

The Jacobian and the Ginzburg-Landau energy

We study properties of Ginzburg--Landau functionals $I^\e_U(\cdot)$, defined for functions $u\in W^{1,n}(U; {\cal R}^n)$, where $U\subset {\cal R}^n$. In particular, we establish lower bounds relating the energy $I^\e_U(u)$ to the Brouwer degree of u, and we prove under additional hypotheses that the energy concentrates on a small number of small sets. As a consequence we deduce some compactness theorems. Such estimates are useful in studying Ginzburg--Landau-type PDEs associated with the functional $I^\e_U$.

https://epubs.siam.org/doi/pdf/10.1137/S0036141097300581

Lower bounds for generalized Ginzburg-Landau functionals

https://www.math.purdue.edu/~phillips/MA694/Jerrard-Soner.pdf

Dynamics of Ginzburg‐Landau Vortices

We study the semiclassical limit of subcritical focussing NLS with a potential, for initial data of the form s(x−x0 )e v0·x , where s is the ground state of an associated unscaled problem. We show that in the semiclassical limit, the solution has roughly the form s( x−x (t) )e v (t)·x , and we show that the approximate center of mass x (·) converges to a solution of the equation x′′ = −DV (x), x(0) = x0, x′(0) = v0 as → 0.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2000/0007/0003/MRL-2000-0007-0003-a007.pdf

Robert L. Jerrard

Papers

The Jacobian and the Ginzburg-Landau energy

Lower bounds for generalized Ginzburg-Landau functionals

Dynamics of Ginzburg‐Landau Vortices

Soliton dynamics in a potential

Vortex dynamics for the Ginzburg-Landau-Schrodinger equation