We consider the 2-dim water wave problem -- the free boundary problem of the Euler equation with gravity and possibly surface tension -- of finite depth linearized at a uniformly monotonic shear flow \(U(x_2)\). Our main focuses are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes (i.e. embedded eigenvalues), we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain stronger decay for the remainders. The linearized gravity waves will also be discussed briefly if time permits. This is a joint work with Xiao Liu.

The work is devoted to the studies of the existence of
solutions of an integro-differential equation in the case of the double
scale anomalous diffusion with the sum of the two negative Laplacians
raised to two distinct fractional powers in \(\mathbb{R}^d\), \(d=4,5\). The proof of the
existence of solutions is based on a fixed point technique. Solvability
conditions for the non-Fredholm elliptic operators in unbounded domains
are used.

https://umsystem.zoom.us/j/94101463494?pwd=NDJaR21PUCtVM0tQWUt0YlNFTmw0UT09

Meeting ID: 941 0146 3494
Passcode: 714934

Extending the Sobolev theory of quasiconformal and quasiregular maps to subdomains of the complex plane motivates our investigation of Sobolev regularity of singular integral operators on domains. We introduce new paraproducts which lead to higher order T1-type testing conditions. A special case provides weighted Sobolev estimates for the compressed Beurling transform which imply quantitative Sobolev estimates for the Beltrami resolvent. This is joint work with Francesco Di Plinio and Brett D. Wick.

In this talk we will survey some old and new results on maximum principles for P-functions and their applications to the study of partial differential equations. More precisely, we will show how one can employ the maximum principle in problems of physical or geometrical interest, in order to get the shape of some free boundaries, isoperimetric inequalities, symmetry results, convexity results and Liouville type results. In the first part of the talk  we'll be mainly focused on some overdetermined problems, while in the second part of the talk we'll present our contributions to some Monge-Ampere type problems and eventually discuss some open problems.

The problem of computing invariants of natural stacks of curves has a long history, starting from Mumford's seminal paper on the Picard group of the stack of 1-pointed elliptic curves. The Picard group of the stack \(\mathcal{M}_{g,n}\) of \(n\)-pointed smooth curves of genus \(g\geq3\) was later computed over \(\mathbb{C}\) by Harer.

We study the closed substack \(\mathcal{H}_{g,n}\) in \(\mathcal{M}_{g,n}\) of \(n\)-pointed smooth hyperelliptic curves of genus \(g\), and compute its Picard group. As a corollary, taking \(g=2\) and recalling that \(\mathcal{H}_{2,n}=\mathcal{M}_{2,n}\), we obtain \(\mathrm{Pic}(\mathcal{M}_{2,n})\) for all \(n\).

Moreover, we give a very explicit description of the generators of the Picard group, which have evident geometric meaning.

We discuss various results on singular integrals adapted to entangled dilations from the past two years. The existing results are mostly on the so-called Zygmund dilations that constitute the simplest intermediate dilation structure lying in between the classical one-parameter setting and the multi-parameter setting. We start with an overview of the subtle optimal weighted theory in the Zygmund case, the techniques behind that, and the implications these have for further results, such as, commutator estimates.  We then discuss the more recent multilinear versions of this theory, the current limitations and, time permitting, some possible further directions and challenges in the area.

We prove a wavelet $T(1)$ theorem for compactness of multilinear Calder\'{o}n-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.

Motivated by recent result of F. Perez and R.R.G. on equality of test ideal of module closure operation and trace ideal, and the well-known result by K.E. Smith that parameter test ideal can never be contained in parameter ideals, we study the obstruction of containment of trace ideals in ideals of finite projective (or injective) dimension. As consequences of our results , we give upper bounds on m-adic order of trace ideals of certain modules. We also prove analogous results for ideal of entries of maps in a free resolution of certain modules. This is joint work with Souvik Dey.

The Camassa-Holm-Kadomtsev-Petviashvili equation (CH-KP) is a two dimensional generalization of the Camassa-Holm equation which has been recently derived in the context of shallow water waves and nonlinear elasticity. In this talk we will discuss the stability of the one-dimensional traveling waves, solitary or periodic, with respect to two dimensional perturbations which are periodic in the transverse direction. We show that the stability or instability depends on a sign parameter of the transverse dispersion term. In particular, a nonlinear instability of the one-dimensional solitary waves of any size can be proved for the so-called CH-KP-I model, while for one-dimensional periodic waves we are able to obtain spectral instability for small amplitude CH-KP-I waves.  This is a joint work with Lili Fan, Jie Jin, Xingchang Wang and Runzhang Xu.