Speaker
Description
A gradient system (X, ℰ, ℛ) consists of a state space X (a separable, reflexive Banach space), an energy functional ℰ : dom(ℰ) ⊂ X → ℝ ∪ {∞}, and a dissipation potential ℛ : X → [0, ∞), which is convex, lower semicontinuous, and satisfies ℛ(0) = 0. The associated gradient-flow equation is given by
0 ∈ ∂ ℛ(u'(t)) + ∇ ℰ(u(t)) or equivalently u'(t) ∈ ∂ ℛ*(−∇ ℰ(u(t))).
In this talk, we consider the case where the dual dissipation potential ℛ is given by the sum ℛ = ℛ₁ + ℛ₂ for two dissipation potentials ℛⱼ : Xⱼ → [0, ∞), where Xⱼ ⊂ X. This decomposition also leads to a splitting of the right-hand side of the combined gradient-flow equation:
u'(t) ∈ ∂ (ℛ₁* + ℛ₂*)(−∇ ℰ(u(t))) = ∂ ℛ₁*(−∇ ℰ(u(t))) + ∂ ℛ₂*(−∇ ℰ(u(t))).
This enables the construction of solutions via a split-step method.
To do this, let τ = T/N define a uniform partition of the interval [0,T]. Starting from an initial datum u₀ ∈ dom(ℰ), we define a piecewise constant time-discrete solution U_τ : [0,T] → dom(ℰ) ⊂ X₁ by setting U_τ(0) = u₀ and performing the Alternating Minimizing Movement, given by:
U_τ(t) = Uₖ¹ for t ∈ ((k{−}1)τ, (k{−}1/2)τ], U_τ(t) = Uₖ² for t ∈ ((k{−}1/2)τ, kτ],
where Uₖ¹ ∈ Argmin_ {U ∈ X₁} { τ/2 ℛ₁ (2/τ} (U − Uₖ₋₁²)) + ℰ((k−1/2)τ, U)},
Uₖ² ∈ Argmin_ {U ∈ X₂} { τ/2 ℛ₂ (2/τ (U − Uₖ¹) ) + ℰ(kτ, U) }.
In this talk, I will demonstrate how the curves U_τ converge to the solution of the combined gradient-flow equation (involving ℛ = ℛ₁ + ℛ₂*) as N → ∞. The analysis relies on methods from the calculus of variations and the use of the energy-dissipation principle for gradient flows.
This talk is based on joint work with Alexander Mielke (Berlin) and Riccarda Rossi (Brescia).
