Dynamics of the Schrödinger–Langevin equation

Abstract

We solve the Schrödinger equation with logarithmic nonlinearity and multiplicative spatial white noise on R d with d ≤ 2. Because of the nonlinearity, the regularity structures and the paracontrolled calculus can not be used. To solve the equation, we rely on an exponential transform that has proven useful in the context of other singular SPDEs.

Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation

“…In the mathematics community, this equation has not seen much interest yet, despite the presence of unusual nonlinear effects. In [10], the author has shown that in the focusing case (λ < 0) every well-prepared density function of the solution of equation (1.2) on R d converges to a Gaussian function weakly in L 1 (R d ), whose mass and center are uniquely determined by its initial data ρ 0 and u 0 , whereas in the defocusing case (λ > 0) every solution disperses to 0, with a slower dispersion rate than usual directly affected by the nonlinear Langevin potential 1 2i log (ψ/ψ * ). Still in the defocusing case, up to a space-time rescaling incorporating dispersive effects, the density ρ of every rescaled solution of equation (1.2) on R d also converges to a Gaussian function weakly in L 1 (R d ), a phenomenon which is reminiscent of the defocusing logarithmic Schrödinger equation [9] (corresponding to the case µ = 0 in (1.2)).…”

The isothermal limit for the compressible Euler equations with damping

“…Of course, our notion of global weak solution has to be adapted to the d-dimensional case, as well as system (1.3), and we refer to [6] and [8] for some d-dimensional analogue of Definition 2.1.…”

“…We recall (see [8]) that there exists a unique global solution τ ∈ C ∞ ([0, ∞)) to this nonlinear ODE, and that this solution remains uniformly bounded from below by a strictly positive constant. Plugging these expressions into the second equation of (3.2), we also get that…”

“…In order to know the behavior of our Gaussian functions when t → ∞, we only need to get some equivalents of the real function τ , which is the goal of the following proposition. We refer to [8] for a complete study of the differential equation (3.4). Proposition 3.2.…”

“…as we actually know from [8,Proposition 3.11] that there exists a time T > 0 such that, for all t ≥ T , 2…”