Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schr\"odinger system

We study a nonlinear system of partial differential equations in which a complex field (the Higgs field) evolves according to a nonlinear Schroedinger equation, coupled to an electromagnetic field whose time evolution is determined by a Chern-Simons term in the action. In two space dimensions, the Chern-Simons dynamics is a Galileo invariant evolution for A, which is an interesting alternative to the Lorentz invariant Maxwell evolution, and is finding increasing numbers of applications in two dimensional condensed matter field theory. The system we study, introduced by Manton, is a special case (for constant external magnetic field, and a point interaction) of the effective field theory of Zhang, Hansson and Kivelson arising in studies of the fractional quantum Hall effect. From the mathematical perspective the system is a natural gauge invariant generalization of the nonlinear Schroedinger equation, which is also Galileo invariant and admits a self-dual structure with a resulting large space of topological solitons (the moduli space of self-dual Ginzburg-Landau vortices). We prove a theorem describing the adiabatic approximation of this system by a Hamiltonian system on the moduli space. The approximation holds for values of the Higgs self-coupling constant close to the self-dual (Bogomolny) value of 1. The viability of the approximation scheme depends upon the fact that self-dual vortices form a symplectic submanifold of the phase space (modulo gauge invariance). The theorem provides a rigorous description of slow vortex dynamics in the near self-dual limit.Comment: Minor typos corrected, one reference added and DOI give

Conservation laws arising in the study of forward-forward Mean-Field Games

We consider forward-forward Mean Field Game (MFG) models that arise in numerical approximations of stationary MFGs. First, we establish a link between these models and a class of hyperbolic conservation laws as well as certain nonlinear wave equations. Second, we investigate existence and long-time behavior of solutions for such models

Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging

Young measure solutions for a nonlinear parabolic equation of forward-backward type

Abstract: "The scope is to study the nonlinear parabolic problem of forward-backward type u[subscript t] = [delta]·q([delta]u) on Q[subscript infinity] = [omega] x R⁺ with initial data u₀ given in H¹₀([omega]). Here [omega] [contained within] R[superscript N] is open, bounded with mildly smooth boundary and q [element of] C(R[superscript N];R[superscript N]), an analogue to heat flux, satisfies q = [delta phi] with [phi element of] C¹ (R[superscript N]) of suitable growth. When [phi] is not convex classical solutions do not exist in general; the problem admits Young measure solutions. By that is meant a function u [element of] H¹[subscript loc](Q[subscript infinity]) [intersection of] L[superscript infinity] (R⁺; H¹₀(omega)) and a parameterized family of probability measures v = (v[subscript x,t])(x,t)[element of]Q[subscript infinity] related to u by [delta]u = fR[superscript N][lambda v][d[lambda]) a.e. in Q[subscript infinity]; via v the nonlinearity q([delta]u)is replaced by the moment = fR[superscript N]q(lambda])v(d[lambda]) a.e. in Q[subscript infinity] and the equation is then interpreted in H[superscript -1]. The family v is generated by the gradients of a sequence in H¹[subscript loc](Q[subscript infinity]), is non-unique, but through its first moment some of the classical properties are preserved: uniqueness of the function u is true; stability is reflected in a maximum principle and a comparison result. The asymptotic analysis yields, as time tends to infinity, a unique limit z and an associated Young measure v[superscript infinity] such that the pair (Z,v[superscript infinity]) is a Young measure solution of the steady-state problem [delta] · q([delta]z) = 0. The relevant energy function is shown to be monotone decreasing and asymptotically tending to its minimum, globally and locally in space.