Fixed points of a destabilized Kuramoto–Sivashinsky equation

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

“…This evolution equation is based on the gradient in the H −1 -topology and uses homogeneous Neumann boundary conditions for both u and ∆u. The diblock copolymer model (2) is an extension of the celebrated Cahn-Hilliard equation [6], which corresponds to the special case σ = 0 and serves as a fundamental model for the phase separation phenomena spinodal decomposition [31,32,44,45,52] and nucleation [3,4,18]. The quantity µ average of the solution over the domain is conserved in time.…”

“…But how does this multistability manifest itself in the evolution equation (2)? Based on the underlying physical situation, one is usually interested in studying the long-term behavior of solutions of the diblock copolymer model which originate close to the (unstable) homogeneous equilibrium state u ≡ µ.…”

“…In each diagram, the vertical axis measures the L 2 (0, 1)-norm of the solutions, and the horizontal axis uses the parameter λ = 1/ 2 . Each colored point corresponds to at least one stationary solution of the diblock copolymer model, and the color indicates the Morse index of the equilibrium in the evolution equation (2), as described in the figure caption. The solution branches are color-coded by the Morse index of the solutions, and black, red, blue, green, magenta, and cyan correspond to indices 0, 1, 2, 3, 4, and 5, respectively.…”

“…Our main result is the following theorem Theorem 1.1 (Existence of heteroclinic connections). Consider the diblock copolymer equation (2) on the one-dimensional domain Ω = (0, 1), for interaction lengths λ = 1/ 2 = 16π 2 and σ = 16, and for total mass µ = 0. Then there exist heteroclinic connections between the unstable homogeneous stationary state u ≡ µ and each of the two local energy minimizers which are indicated by yellow circles in Figure 2, and shown in the top right panel of Figure 3.…”

“…In addition, there exist heteroclinic connections between the unstable homogeneous stationary state and each of the two suspected global energy minimizers which are indicated by yellow stars in Figure 2, and which are shown in the top left panel of Figure 3. In other words, for the above parameter values the diblock copolymer equation (2) exhibits multistability in the sense that local or suspected global energy minimizers can be reached from the homogeneous state.…”