Author Archives: federico

Kihyun Kim (KAIST)

The APDE seminar on Monday, 2/8, will be given by Kihyun Kim online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation

Abstract: We consider the blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariance symmetry. (CSS) is $L^2$-critical, has the pseudoconformal symmetry, and admits a soliton $Q$ for each equivariance index $m \geq 0$. An application of the pseudoconformal transformation to $Q$ yields an explicit finite-time blow-up solution $S(t)$ which contracts at the pseudoconformal rate $|t|$. In the high equivariance case $m \geq 1$, the pseudoconformal blow-up for smooth finite energy solutions in fact occurs in a codimension one sense, but also exhibits an instability mechanism. In the radial case $m=0$, however, $S(t)$ is no longer a finite energy blow-up solution. Interestingly enough, there are smooth finite energy blow-up solutions whose blow-up rates differ from the pseudoconformal rate by a power of logarithm. We will explore these interesting blow-up dynamics (with more focus on the latter) via modulation analysis. This talk is based on my joint works with Soonsik Kwon and Sung-Jin Oh.

Khang Manh Huynh (UCLA)

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.

Shuang Miao (Wuhan University)

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.

Jérémie Szeftel (UPMC)

The APDE seminar on Monday, 10/12, will be given by Jérémie Szeftel 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: Finite time blow up for focusing supercritical NLS and compressible fluids

Abstract: I will present recent results in collaboration with Frank Merle, Pierre Raphaël and Igor Rodnianski concerning finite time blow up for the focusing supercritical NLS equation, for compressible Euler equations, and for compressible Navier-Stokes equations.