The APDE seminar on Monday, 11/30, will be given by Khang Manh Huynh online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Construction of the Hodge-Neumann heat kernel, local Bernstein estimates, and Onsager’s conjecture in fluid dynamics.

Abstract: Most recently, in arXiv:1907.05360 [math.AP], we introduced the theory of heatable currents and proved Onsager’s conjecture on Riemannian manifolds with boundary, where the weak solution has $B_{3,1}^{\frac{1}{3}}$ spatial regularity. In this sequel, by applying techniques from geometric microlocal analysis to construct the Hodge-Neumann heat kernel, we obtain off-diagonal decay and local Bernstein estimates, and then use them to extend the result to the Besov space $\widehat{B}_{3,V}^{\frac{1}{3}}$, which generalizes both the space $\widehat{B}_{3,c(\mathbb{N})}^{1/3}$ from arXiv:1310.7947 [math.AP] and the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from arXiv:1902.07120 [math.AP] — the best known function space where Onsager’s conjecture holds on flat backgrounds.

The APDE seminar on Monday, 11/23, will be given by Grigorios Fournodavlos online via Zoom from 9:10 to 10am (note the time change). To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Asymptotically Kasner-like singularities.

Abstract: The Kasner metric is an exact solution to the Einstein vacuum
equations, containing a Big Bang singularity. Examples of more general
singularities in the vicinity of Kasner are in short supply, due its
complicated dynamics. I will present a recent joint work with Jonathan
Luk, which constructs a large class of singular solutions with
Kasner-like behavior, without symmetry or analyticity assumptions.

The APDE seminar on Monday, 11/16, will be given by Shuang Miao online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: The stability of blow up solutions to critical Wave Maps beyond equivariant setting

Abstract: In 2006, Krieger, Schlag and Tataru (KST) constructed a family of type II blow up solutions to the 2+1 dimensional wave map equation with unit sphere as its target. This construction provides the first example of blow up solutions to the energy-critical Wave Maps. A key feature of this family is that it exhibits a continuum of blow up rates. However, from the way it was constructed, the stability of this family was not clear and it was believed to be non-generic. In this talk I will present our recent work on proving the stability and rigidity of the KST family, beyond the equivariant setting. This is based on joint works with Joachim Krieger and Wilhelm Schlag.