We prove that the twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the annulus, is stable to perturbations. In fact, it is possible to prove the stability of the whole of the lifted dynamics to non-autonomous perturbations, though single particle paths are generically unstable. These all-time stability facts are used to establish a number of results related to the long-time behavior of fluid flows. (Joint work with T. Drivas and T. Elgindi)

In this lecture series, I will discuss rigidity in the long time dynamics of some evolution equation. The specific equation to be paid attention to is the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. (CSS) is a gauged 2D cubic nonlinear Schrödinger equation that has pseudoconformal invariance and solitons. However, two distinguished features of (CSS), the self-duality and non-locality, make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We will explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem, in detail.

In this lecture series, I will discuss rigidity in the long time dynamics of some evolution equation. The specific equation to be paid attention to is the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. (CSS) is a gauged 2D cubic nonlinear Schrödinger equation that has pseudoconformal invariance and solitons. However, two distinguished features of (CSS), the self-duality and non-locality, make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We will explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem, in detail.

We consider the nonlinear Schrödinger equation and the nonlinear wave equation with initial data decaying slower than L^2 functions. However, the L^p-spaces for p \neq 2 are not invariant under the linear propagation. We consider function spaces, which allow for decay like in L^p, p > 2, and which are invariant under the linear propagation. We show L^p-smoothing estimates using \ell^2-decoupling due to Bourgain-Demeter. The results on nonlinear wave equations are joint work with Jan Rozendaal (IMPAN).

In this lecture series, I will discuss rigidity in the long time dynamics of some evolution equation. The specific equation to be paid attention to is the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. (CSS) is a gauged 2D cubic nonlinear Schrödinger equation that has pseudoconformal invariance and solitons. However, two distinguished features of (CSS), the self-duality and non-locality, make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We will explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem, in detail.