We analytically and numerically study a fourth-order PDE modeling rough crystal surface diffusion on the macroscopic level. We discuss existence of solutions globally in time and long-time dynamics for the PDE model. The PDE, originally derived by Katsevich is the continuum limit of a microscopic model of the surface dynamics, given by a Markov jump process with Metropolis-type transition rates. We outline the convergence argument, which depends on a simplifying assumption on the local equilibrium measure that is valid in the high-temperature regime. We provide numerical evidence for the convergence of the microscopic model to the PDE in this regime.

We consider the motion of several rigid bodies immersed in a two-dimensional incompressible perfect fluid, the whole system occupying a bounded simply connected domain. The external fixed boundary is impermeable except on an open nonempty part where one controls both the normal velocity, allowing some fluid to go in and out of the domain, and the entering vorticity. The motion of the rigid bodies is given by the Newton laws with forces due to the fluid pressure and the fluid motion is described by the incompressible Euler equations. We prove that it is possible to exactly achieve any noncolliding smooth motion of the rigid bodies by the remote action of a controlled normal velocity on the outer boundary which takes the form of state-feedback, with zero entering vorticity. The proof relies on a nonlinear method to solve linear perturbations of nonlinear equations associated with a quadratic operator.

We consider the Fröhlich Hamiltonian with large coupling constant $\alpha$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm-approximation of the evolution, valid up to times of order ${\alpha }^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau–Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau–Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order ${\alpha }^2$.

We consider a system of $N$ interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true $N$-body dynamics in norm to arbitrary precision. The $N$-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick’s theorem. We determine the $n$-point correlation functions of the excitations around the condensate, as well as the reduced densities of the $N$-body system, to arbitrary accuracy, given only the knowledge of the two-point correlation functions of a quasifree state and the solution of the Hartree equation. In this way, the complex problem of computing all $n$-point correlation functions for an interacting $N$-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point correlation functions.

A nonlinear parabolic equation of sixth order is analyzed. The equation arises as a reduction of a model from quantum statistical mechanics and also as the gradient flow of a second-order information functional with respect to the $L^2$-Wasserstein metric. First, we prove global existence of weak solutions for initial conditions of finite entropy by means of the time-discrete minimizing movement scheme. Second, we calculate the linearization of the dynamics around the unique stationary solution, for which we can explicitly compute the entire spectrum. A key element in our approach is a particular relation between the entropy, the Fisher information and the second-order functional that generates the gradient flow under consideration.

We study the effects of dispersive stabilization on the compressible Euler equations in Lagrangian coordinates in the one-dimensional torus. We assume a van der Waals pressure law, which presents both hyperbolic and elliptic zones. The dispersive stabilization term is of Schrödinger type. In particular, the stabilized system is complex-valued. It has a conservation law, which, for real unknowns, is identical to the energy of the original physical system. The stabilized system supports high-frequency solutions, with an existence time or an amplitude which depend strongly on the pressure law.

The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the earth can be in principle reconstructed from single scattering measurements at the surface.