An improved estimate on Fourier coefficients of restricted eigenfunctions via geodesic beams
In this talk, we will discuss the behavior of Laplace eigenfunctions with frequency $\lambda_j$ on a compact, Riemannian manifold when restricted to a submanifold. We analyze the restricted eigenfunctions by studying their Fourier coefficients with respect to an orthonormal basis of eigenfunctions on the submanifold with frequency $\mu_k$. We consider a particular sum of the norm-squares of these Fourier coefficients over joint frequencies $\{(\lambda_j,\mu_k)\}$ in a suitably thick strip. By imposing a condition on the dynamics of the geodesic flow, we obtain a logarithmic improvement on the error term of the asymptotics of this sum obtained by Wyman, Xi, and Zelditch. This improvement allows us to better understand the average behavior of the Fourier coefficients in a smaller window around the frequency $\lambda_j$.
On the well-posedness of the KP-I equation on R2
We survey some recent results on the low-regularity and unconditional well-posedness for KP-I equation posed on R2.
Well-posedness of dispersion generalized KP-I equations
In this talk, we consider the Cauchy problem of dispersion-generalized KP-I equations. It is known that the well-posedness of the original KP-I equation cannot be obtained via fixed-point argument in any anistropic Sobolev space. We obtain the almost sharp well-posedness results via iteration in anistropic Sobolev spaces when dispersion is large enough. We use the standard Fourier restriction norm method. The key tool is the convolution estimate for functions supported on thickened hypersurfaces. This talk is based on a joint work with Akansha Sanwal and Robert Schippa.
Continuum Calogero--Moser models
The focusing CCM model is a dispersive equation that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential. Recently, Gérard and Lenzmann discovered solutions to this equation that exhibit turbulent behavior.
In this talk, we will discuss a scaling-critical well-posedness result for the focusing and defocusing CCM models on the line. In the focusing case, this requires solutions to have mass less than that of the soliton. This is joint work with Rowan Killip and Monica Visan.
Relativistic Fluids in Expanding Cosmologies
The FLRW solution is the simplest cosmological model in general relativity, describing a fluid-filled, spatially homogeneous universe. There is extensive literature in the physics community on cosmological models with a linear equation of state p=K\rho, however rigorous mathematical results, until recently, have been confined to perturbations of FLRW with K≤1/3. The simplest generalisation of the FLRW model is the Bianchi class of solutions, which allow for spatial anisotropy. I will discuss joint work with G. Fournodavlos and T. Oliynyk proving the non-linear future stability of Bianchi solutions for K>1/3. Additionally, I will present numerical work, with F. Beyer and T. Oliynyk, on the future instability of FLRW models with super-radiative equations of state.
Preprint 1 and preprint 2
On the asymptotic behavior of cubic NLS systems without weak null condition
In this talk, we discuss cubic NLS systems in one dimension (1D). It is well-known that cubic nonlinearity is critical in 1D, where nonlinear effects significantly influence the asymptotic behavior of small solutions (e.g., modified scattering). In the case of systems, these nonlinear effects exhibit a wide variety of phenomena.
Previous studies in this field often assume the weak null condition, a structural condition that ensures the existence of an effective conserved quantity. In this work, we present an example of a system that does not satisfy this condition. Despite this, we succeed in determining the asymptotic behavior of small solutions. The key lies in identifying a new type of conserved quantity for the corresponding ODE system.
Global dynamics around multi-solitons for the nonlinear Klein-Gordon equation
We consider the nonlinear Klein-Gordon equation with focusing cubic power in three dimensions, and investigate global behavior of solutions for initial data close to a superposition of solitons. Such a form is expected as asymptotic behavior for large time of generic global solutions by the soliton resolution conjecture, but its relation to the initial data is highly nontrivial since the solitons are unstable. I will explain how to connect the linearized analysis of the instability to the global behavior of blow-up and scattering, as well as to the initial data and their classification by invariant manifolds.
Decay for the Chern-Simons-Higgs equations
In this talk, I will present a joint work with D. Wei on the long time dynamics for solutions to the Chern-Simons-Higgs equation with a pure power defocusing nonlinearity in two space dimension. We show that the potential energy decays inverse polynomially in time. Sharp pointwise decay estimate also holds for sufficiently large power of nonlinearity. The proof relies on vector field method and a sharp geometric trace theorem developed by Klainerman-Rodnianski.