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We study the stability of a so-called kink profile for the one-dimensional Cahn-Hilliard problem on the real line. We derive optimal bounds on the decay to equilibrium under the assumption that the initial energy is less than three times the energy of a kink and that the initial H −1 distance to a kink is bounded. Working with theḢ −1 distance is natural, since the equation is a gradient flow with respect to this metric. Indeed, our method is to establish and exploit elementary algebraic and differential relationships among three natural quantities: the energy, the dissipation, and theḢ −1 distance to a kink. Along the way it is necessary and possible to control the timedependent shift of the center of the L 2 closest kink. Our result is different from earlier results because we do not assume smallness of the initial distance to a kink ; we assume only boundedness.

A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple one-dimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson-Mehl-Avrami-Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.

We establish metastability of the one-dimensional Cahn-Hilliard equation for initial data that is order-one in energy and order-one inḢ −1 away from a point on the so-called slow manifold with N well-separated layers. Specifically, we show that, for such initial data on a system of lengthscale Λ, there are three phases of evolution: (1) the solution is drawn after a time of order Λ 2 into an algebraically small neighborhood of the N-layer branch of the slow manifold, (2) the solution is drawn after a time of order Λ 3 into an exponentially small neighborhood of the N-layer branch of the slow manifold, (3) the solution is trapped for an exponentially long time exponentially close to the N-layer branch of the slow manifold. The timescale in phase (3) is obtained with the sharp constant in the exponential.

We consider a hybrid compressible/incompressible system with memory effects, introduced recently by Lefebvre Lepot and Maury for the description of one-dimensional granular flows. We prove a global existence result for this system without assuming additional viscous dissipation. Our approach extends the one by Cavalletti et al. for the pressureless Euler system to the constrained granular case with memory effects. We construct Lagrangian solutions based on an explicit formula using the monotone rearrangement associated to the density. We explain how the memory effects are linked to the external constraints imposed on the flow. This result can also be extended to a heterogeneous maximal density constraint depending on time and space.