The APDE seminar on Monday, 11/1, will be given by Sung-Jin Oh (UC Berkeley) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: A tale of two tails

Abstract: In this talk, I will introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, the method also applies to dynamical spacetimes. In this case, I will explain how the late-time tails are in general different(!) from the stationary case in the presence of dynamical and/or nonlinear perturbations of problem. This is joint work with Jonathan Luk (Stanford).

The APDE seminar on Monday, 10/25, will be given by Junyan Zhang (Johns Hopkins University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu)

Title: Anisotropic regularity of the free-boundary problem in ideal compressible MHD

Abstract: We consider the free-boundary compressible ideal MHD system under the Rayleigh-Taylor sign condition. The local well-posedness was recently proved by Trakhinin and Wang by using Nash-Moser iteration. We prove the a priori estimate without loss of regularity in the anisotropic Sobolev space. Our proof is based on the combination of the “modified” Alinhac good unknown method, the full utilization of the structure of MHD system and the anisotropy of the function space. This is the joint work with Professor Hans Lindblad.

The APDE seminar on Monday, 10/18, will be given by our own Michael Christ online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: On quadrilinear implicitly oscillatory integrals

Abstract: Multilinear oscillatory integrals arise
in various contexts in harmonic analysis,
in partial differential equations, in ergodic theory,
and in additive combinatorics.  We discuss the majorization of integrals
$\int \prod_{j} (f_j\circ\varphi_j)$ of finite products
by negative order norms of the factors,
where integration is over a ball in Euclidean space and $\varphi_j$ are smooth
mappings to a space of strictly lower dimension. The talk focuses
on the quadrilinear case, after work on the trilinear case of Bourgain (1988),
of Joly, M\’etivier, and Rauch (1995), and of the speaker (2019).
Sublevel set inequalities, which quantify the nonsolvability of certain systems
of linear equations, are a central element of the analysis.

The APDE seminar on Monday, 10/11, will be given by Jacek Jendrej (Université Sorbonne Paris Nord) online via Zoom from 9.10am to 10.00am PST (note the time change). To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu)

Title: Soliton resolution for energy-critical equivariant wave maps

Abstract: We consider wave maps R^(1+2) -> S^2, under the assumption of equivariant symmetry. We prove that every solution of finite energy resolves, as time passes, into a superposition of harmonic maps (solitons) and radiation. It was proved in works of Côte, and Jia and Kenig, that such a decomposition holds along a sequence of times. We show that the resolution holds continuously in time via a “no-return lemma” based on the virial identity. The proof combines a modulation analysis of solutions near a multi-soliton configuration with the concentration-compactness method. Joint work with Andrew Lawrie from MIT.