# Graduate seminar on Advanced topics in PDE

## Organizers

• Prof. Dr. Herbert Koch
• Prof. Dr. Christoph Thiele
• Dr. Alex Amenta
• João Pedro Ramos
• ## Schedule

This seminar takes place regularly on Fridays, at 14.00 (c.t.), in Raum 0.011, Endenicher Allee 60.

##### Title:
Banach-valued modulation-invariant Carleson embeddings and outer measure spaces: the Walsh case

##### Abstract:
Consider three Banach spaces $X_0, X_1, X_2$, linked with a bounded trilinear form $\Pi : X_0 \times X_1 \times X_2 \to \mathbb{C}$. Given this data one can define Banach-valued analogues of the bilinear Hilbert transform and its associated trilinear form. Using the Do-Thiele theory of outer $L^p$-spaces, $L^p$-bounds for these objects can be reduced to modulation-invariant Carleson embeddings of $L^p(\mathbb{R};X_v)$ into appropriate outer $L^p$-spaces. We prove such embeddings in the Banach-valued setting for a discrete model of the real line, the 3-Walsh group. Joint work with Gennady Uraltsev (Cornell).

##### Title:
On the well-posedness for higher order parabolic equations with rough coefficients

##### Abstract:
In the first part we study the existence and uniqueness of solutions to the higher order parabolic Cauchy problems on the upper half space, given by $\partial_t u = (-1)^{m+1} \mbox{div}_m A(t,x)\nabla^m u$ and $L^p$ initial data space. The (complex) coefficients are only assumed to be elliptic and bounded measurable. Our approach follows the recent developments in the field for the case $m=1$. In the second part we consider the $BMO$ space of initial data. We will see that the Carleson measure condition $$\sup_{x\in \mathbb{R}^n} \sup_{r>0} \frac{1}{|B(x,r)|}\int_{B(x,r)}\int_0^{r}|t^m\nabla^m u(t^{2m},x)|^2\frac{dxdt}{t}<\infty$$ provides, up to polynomials, a well-posedness class for $BMO$. In particular, since the operator $L$ is arbitrary, this also leads to a new, broad Carleson measure characterization of $BMO$ in terms of solutions to the parabolic system.

##### Title:
Bi-parameter Carleson embedding

##### Abstract:
Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on bi-tree. In this note we give one more proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree that used the Bellman function technique, the proof here is based on some rather subtle comparison of energies of measures on bi-tree. The bi-tree Carleson embedding theorem turns out to be very different from the usual one on a simple tree. In particular, various types of Carleson conditions are not equivalent in general for bi-parameter case.

##### Title:
Broken Symmetries of the Schrodinger Equation and Strichartz Estimates

##### Abstract:
This will be short talk where we report some of the partial results of an ongoing work with Don Zagier. We study the Schrodinger equation from the point of view of Hermite and Laguerre expansions and establish a diagonalization result for initial data with prescribed parity in 3 dimensions that present exotic and unexpected associated eigenvalues. In particular, we derive a sharpened inequality for the one dimensional Strichartz inequality for even initial data. For odd initial data we prove the extremizer is the derivative of a Gaussian. We remark this is still unfinished work, and some questions are still left to be answered, so the audience is more than welcome to ask all sorts of questions.

##### Title:
Blowup stability for wave equations with power nonlinearity

##### Abstract:
We introduce the method of similarity coordinates to study the stability of ODE blowup solutions of wave equations with power nonlinearity in the lightcone. We first recall stability results in higher Sobolev spaces. In this case, using the Lumer--Philips theorem we obtain a solution semigroup to the Cauchy problem. Then by the Gearhart--Prüss theorem we obtain enough decay of the semigroup to control the nonlinearity. Then we show stability of the ODE blowup for the energy critical equation in energy space, by establishing Strichartz estimates in similarity coordinates. In this case the Gearhart--Prüss theorem does not give a useful bound. Hence we need to construct an explicit expression of the semigroup, from which we are finally able to prove Strichartz estimates and an improved energy estimate to control the nonlinearity in the energy space. The result in the energy critical case in $d=5$ is by Roland Donninger and myself (the pioneering work of the $d=3$ case is by Roland Donninger).

##### Title:
Concentration inequalities for bandlimited functions

##### Abstract:
This talk considers the following problem: How much of an integral norm of a function with compactly supported Fourier transform can be concentrated on a sparse set? The resulting inequalities have explicit bounds that depend on the size of the support of the transform and on a measure of sparsity of the set. I will describe some applications from analytic number theory, signal processing, and Lagrange interpolation, and will outline existing strategies (building on work of Selberg and of Donoho and Logan) to obtain concentration inequalities.