Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure

Abstract

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the L 2 -Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.

Well-posedness of evolution equations with time-dependent nonlinear mobility: A modified minimizing movement scheme

“…Clearly, we also do not have to require finiteness of second moments m 2 (u) for probability measures u ∈ P(Ω) on Ω explicitly anymore. However, due the appearance of boundary terms, we need the additional isotropy condition (compare to [18]).…”

“…Specifically, we prove the existence of nonnegative weak solutions to (1.1) using a modified version of the classical minimizing movement scheme for gradient flows (which has been employed for various evolution equations with metric gradient flow structure, e.g. [11,21,1,10,19,12,17,27,29,5], also with spatially varying coefficients [22,15,18]): Definition 1.2 (Modified minimizing movement scheme). Define the space X := u ∈ L 1 (R d ) : 0 ≤ u(x) a.e.…”

Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type

“…The proof of existence of weak solutions below follows the same main lines as in our recent work [35,24]. The backbone is the variational minimizing movement scheme for gradient flows [9,15,2]: For a given step size τ > 0, define a sequence (u k τ ) k≥0 recursively by u 0 τ := u 0 ,…”

“…The presented strategy of proof counts by now as "classical" in the context of gradient flows in the L 2 -Wasserstein distance: after the seminal contributions by Jordan, Kinderlehrer and Otto [15] and Otto [27] on second order equations, it has also been employed e.g. for fourth-order equations [12,13,25,24], equations of fractional order [21], and systems [18,4,6,31,5,16,32]. For the distances with nonlinear mobility, existence results have been derived by this method e.g.…”

The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances

“…if f (u) ∈ H 1 (Ω), and F (u) := +∞ otherwise. We call F the generalized Fisher information functional associated to the mobility m. For linear mobility, functionals of the form (5) for other choices of f have already been studied in [11].…”