Author Archives: mihaela

Marta Lewicka (University of Pittsburgh)

Speaker: Marta Lewicka

Title: “Convex integration for the Monge-Ampere equation in two dimensions”.

 

Abstract:

We discuss the dichotomy of rigidity vs. flexibility for the $\mathcal{C}^{1,\alpha}$ solutions to the Monge-Ampere equation in two dimensions:

\begin{equation}

{\mathcal{D}et} \nabla^2 v := -\frac 12 \mbox{curl curl } (\nabla v \otimes \nabla v) = f \qquad \mbox{in } \Omega\subset\mathbb{R}^2.

\end{equation}

Firstly, we show that below the regularity threshold $\alpha<1/7$, the very weak $\mathcal{C}^{1,\alpha}(\bar\Omega)$ solutions to  the equation above, (\ref{MA}), are dense in the set of all continuous functions.

This flexibility statement is a consequence of the convex integration $h$-principle, whereas we directly adapt the iteration method of Nash and Kuiper in order to construct the oscillatory solutions.

Secondly, we prove that the same class of very weak solutions fails the above flexibility in the regularity regime $\alpha>2/3$.

Our interest in the regularity of Sobolev solutions to the Monge-Ampere equation is motivated by the variational description of shape formation, which I will also explain in the talk.

 

Mihaela Ifrim (April 27th)

Speaker: Mihaela Ifrim (UC Berkeley)

Title: Long time solutions for two dimensional water waves

Abstract: This is joint work with Daniel Tataru, and in parts with John Hunter. My talk is concerned with the infinite depth water wave equation in two space dimensions, with either gravity or surface tension. Both cases will be discussed in parallel. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data.  For the gravity water waves there are several results available; they have been recently obtained by Wu, Alazard-Burq-Zuily and Ionescu-Pusateri using different coordinates and methods. In the capillary water waves case, we were the first to establish a global result.  Our goal is improve the understanding of these problems by providing a single setting for both cases, and  presenting simpler proofs. The talk will be as self contained as the time permits.

Michal Wrochna (April 20th)

 

Speaker: Michal Wrochna (Stanford University)

Title: Scattering theory approach to the Feynman problem for the wave equation

Abstract: A classical result of Duistermaat and Hörmander provides four parametrices for the wave equation, distinguished by their wave front set. In applications in Quantum Field Theory one is interested in constructing the corresponding exact inverses, satisfying in addition a positivity condition. I will present a method (derived in a joint work with C. Gérard and dating back to W. Junker), where this is achieved by diagonalizing the wave equation in terms of elliptic pseudodifferential operators and solving the Cauchy problem with possible smooth remainders. I will then indicate possible ways of replacing Cauchy data by scattering data and comment on how this relates to global constructions of Feynman propagators.

Vlad Vicol (Aprin 6th)

 

Speaker: Vlad Vicol (Princeton University)

Title: Holder continuous solutions of active scalar equations

Abstract: We consider active scalar equations $\partial_t \theta + \nabla \cdot (u\, \theta) = 0$, where $u = T[\theta]$ is a divergence-free velocity field, and $T$ is a Fourier multiplier operator. We prove that when $T$ is not an odd multiplier, there are nontrivial, compactly supported solutions weak solutions, with Holder regularity $C^{1/9-}_{t,x}$. In fact, every integral conserving scalar field can be approximated in $D’$ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when $T$ is odd, weak limits of solutions are solutions, so that the $h$-principle for odd active scalars may not be expected. This is a joint work with Phillip Isett (MIT).

Walter Strauss ( March 30st)

Speaker: Walter Strauss (Brown University)

Title: Stability of a Hot Plasma in a Torus

Abstract: In a tokamak huge numbers of charged particles whiz around a torus
at relativistic speeds. Finding stable particle configurations is
the holy grail of fusion energy research. We model a collisionless
plasma by the relativistic Vlasov-Maxwell system. There are many
equilibria, of which some are stable and some unstable. In this talk I
will present recent work with Toan Nguyen where the particles reflect
specularly and the field is a perfect conductor.  These are however not
the physical boundary conditions. Given an equilibrium of a certain type,
we reduce linear stability to the positivity of a certain non-local linear
operator which is much less complicated than the generator of the full
linearized system.

Tristan Buckmaster (February 32rd)

Speaker: Tristan Buckmaster (NYU)
Title: Onsager’s Conjecture
Abstract: In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy.

The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett and myself related to resolving the second component of Onsager’s conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

Andrew Lawrie (February 9th)

Speaker: Andrew Lawrie (UC Berkeley)
Title: A refined threshold theorem for  $(1+2)$-dimensional wave maps into surfaces. (joint with Sung-Jin Oh)
Abstract:
The recently established threshold theorem of Sterbenz and Tataru for energy critical wave maps states that wave maps with energy less than that of the ground state (i.e., a minimal energy nontrivial harmonic map) are globally regular and scattering on $\mathbb{R}^{1+2}$. In this talk we give a refinement of this theorem when the target is a closed orientable surface by taking into account an additional  invariant of the problem, namely the topological degree. We show that the sharp energy threshold for global regularity and scattering is in fact \emph{twice} the energy of the ground state for wave maps with degree zero, whereas wave maps with nonzero degree necessarily have at least the energy of the ground state.

Jeff Calder (February 2)

Speaker: Jeff Calder (UC Berkeley)

Title: A PDE-proof of the continuum limit of non-dominated sorting

Abstract: Non-dominated sorting is a combinatorial problem that is fundamental in multi-objective optimization, which is ubiquitous in engineering and scientific contexts. The sorting can be viewed as arranging points in Euclidean space into layers according to a partial order. It is equivalent to several well-known problems in probability and combinatorics, including the longest chain problem, and polynuclear growth. Recently, we showed that non-dominated sorting of random points has a continuum limit that corresponds to solving a Hamilton-Jacobi equation in the viscosity sense. Our original proof was based on a continuum variational problem, for which the PDE is the associated Hamilton-Jacobi-Bellman equation. In this talk, I will sketch a new proof that avoids this variational interpretation, and uses only PDE techniques. The proof borrows ideas from the Barles-Souganidis framework for proving convergence of numerical schemes to viscosity solutions. As a result, it seems this proof is more robust, and we believe it can be applied to many other problems that do not have obvious underlying variational principles. I will finish the talk by briefly sketching some current problems of interest.