Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics

Abstract

In this paper, we present a novel center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders. This is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolution problem. Under suitable hypotheses, the resulting center manifold is finite dimensional and captures all sufficiently small bounded solutions. Compared with classical methods, the reduced ODE on the manifold is more directly related to the original physical problem and also easier to compute. The analysis is conducted directly in Hölder spaces, which is often desirable for elliptic equations.We then use this machinery to construct small bounded solutions to a variety of systems. These include heteroclinic and homoclinic solutions of the anti-plane shear problem from nonlinear elasticity; exact slow moving invasion fronts in a two-dimensional Fisher-KPP equation; and hydrodynamic bores with vorticity in a channel. The last example is particularly interesting in that we find solutions with critical layers and distinctive "half cat's eye" streamline patterns.

Large-amplitude steady solitary water waves with constant vorticity

“…By the arguments in [CWW19], this solution is unique. By choosing s 1 , s 2 appropriately we therefore obtain explicit expressions for Υ 200 , Υ 101 , and calculate that ∂ 2…”

“…This would be particularly unpleasant given our elliptic system setting. Recently however, a new center manifold reduction theorem was derived in [CWW19], which we will use instead. On the one hand, this provides us with a comparatively simple method for obtaining the reduced ODE.…”

“…We will prove Theorem 7.1 using the center manifold reduction results in [CWW19]. To state these results, we need weighted versions of the spaces X , Y defined in Section 5.1 allowing for exponential growth, namely…”

“…Proof. We will apply Theorem 1.1 in [CWW19]. This result mostly concerns scalar problems and does not immediately apply to systems of the type (2.11).…”

“…We provide a functional analytic formulation of the problem in Section 5 along with properties of the linearized operator, and in Section 6 we prove our uniform regularity result. In Section 7 we first construct small amplitude solitary wave solutions to our problem using the approach in [CWW19] which allows us, with minimal tedium, to reduce the problem to an ODE. Finally, we construct large-amplitude solutions, thus proving the main theorem of the paper.…”