The APDE seminar on Monday, 4/26, will be given by Rita Teixeira da Costa online via Zoom from 12.10pm to 1.00pm PT (note the time change). To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Oscillations in wave map systems and an application to General Relativity

Abstract: Due to their nonlinear nature, the Einstein equations are not closed under weak convergence. Compactness singulaties associated to highly oscillatory solutions may be identified with some non-trivial matter. In 1989, Burnett conjectured that, for vacuum sequences, this matter produced in the limit is captured by the Einstein-massless Vlasov model. 
In this talk, we give a proof of Burnett’s conjecture under some gauge and symmetry assumptions, improving previous work by Huneau—Luk from 2019. Our methods are more general, and apply to oscillating sequences of solutions to the wave maps equation in (1+2)-dimensions.
This is joint work with André Guerra (University of Oxford).

The APDE seminar on Monday, 4/12, will be given by Jeffrey Kuan online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: A stochastic fluid-structure interaction model given by a stochastic viscous wave equation

Abstract: We consider a stochastic fluid-structure interaction (FSI) model, given by a stochastic viscous wave equation perturbed by spacetime white noise. The wave equation part of the model describes the elastodynamics of a thin structure, such as an elastic membrane, while the viscous part, which is in the form of the Dirichlet-to-Neumann operator, describes the impact of a viscous, incompressible fluid in a two-way coupled fluid-structure interaction problem. The stochastic perturbation describes random deviations observed in real-life data. We prove that this stochastic viscous wave equation has a mild solution in dimension one, and also in dimension two, which is the physical dimension of the FSI problem (thin 2D membrane). This behavior contrasts that of the stochastic heat and the stochastic wave equations, which do not have function valued mild solutions in dimensions two and higher. This means that in the two dimensional model, unlike the heat and wave equations, dissipation due to fluid viscosity in the viscous wave equation, keeps the stochastically perturbed solution “in control”. We also consider Hölder continuity path properties of solutions and show that the solution is Hölder continuous up to Hölder exponent 1/2 in both space and time, after stochastic modification. This is joint work with Suncica Canic.

The APDE seminar on Monday, 3/8, will be given by John Anderson online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: Stability results for anisotropic systems of wave equations

Abstract: In this talk, I will describe a global stability result for a nonlinear anisotropic system of wave equations. This is motivated by studying phenomena involving characteristics with multiple sheets. For the proof, I will describe a strategy for controlling the solution based on bilinear energy estimates. Through a duality argument, this will allow us to prove decay in physical space using decay estimates for the homogeneous wave equation as a black box. The final proof will also require us to exploit a certain null condition that is present when the anisotropic system of wave equations satisfies a structural property involving the light cones of the equations.

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.

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, 11/16, 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.

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.