Since 1993, when the celebrated Camassa-Holm equation was reobtained as a shallow water model with weak peaked solutions, much effort has been put into the study of local and global well-posedness of similar equations with applications in water waves. In this talk we consider a hydrodynamic model that lacks the conservation of energy, a crucial quantity for determination of global well-posedness in Sobolev spaces. We construct a nontrivial functional to estimate the norm of the solution and show that, under certain mild conditions, there is a unique global solution. From that, we then extend regularity of the solution by making use of the powerful machinery of Kato and Masuda and show that it is globally analytic in both variables.

The study of nonlinear lattice dynamical systems is an active area of research of continuously growing interest. Due to their relevance to nonlinear phenomena of restrained energy redistribution, the problem of the existence of discrete localized modes in the form of solitons and breathers in the lattice systems has attracted tremendous interest not only from the physical but also from the rigorous mathematical viewpoint. In this talk, a variety of approaches, ranging from some, to our knowledge, novel topological proofs for the existence of discrete breathers and solitons in Hamiltonian systems, to global bifurcation theory and important tools as the Łojasiewicz-Simon inequality in analyzing the structure of the limit sets of dissipative counterparts will be discussed.

One of the prototypical completely integrable nonlinear evolution equations is the focusing nonlinear Schrödinger equation (NLS), which is a universal model for weakly nonlinear dispersive wave packets, and as such it arises in a variety of physical settings, including deep water, optics, acoustics, condensed matter, etc. Remarkably, solutions of the focusing NLS equation on the circle corresponding to finite-gap initial data can be represented in terms of Riemann theta functions. These so-called finite-gap solutions for focusing NLS arise from the spectral theory of a corresponding Dirac operator. In this talk I will discuss (i) general spectral theory of one-dimensional Dirac operators on the circle, (ii) a particular two-parameter family of Jacobi elliptic potentials, and time permitting (iii) applications to the theory of breather gases in dispersive hydrodynamics.

Initial-boundary value problems (IBVPs) with constant initial and time-dependent boundary data, also known as the wavemaker problem, are fundamental and of significant importance in mathematics and physics. For the linear time-periodic wavemaker problem, the radiation condition selects the unique traveling wave or spatially decaying wave in the neighborhood of the boundary. However, this simple and physically plausible argument has yet to be mathematically justified in a general context. In recent work, the IBVP for some linear and integrable nonlinear evolution equations has been solved using the unified transform method. A related approach, called the Q-equation method, has been introduced to derive the Dirichlet-to-Neumann (D-N) map for asymptotically time-periodic boundary conditions. This talk will extend and prove the existence of the unique D-N map for a general third-order wave model and prove the radiation condition as a consequence. In addition, two representative linear evolution equations with sinusoidal boundary conditions are studied and uniform asymptotic approximations are obtained for large $t$, one of which is shown to give a quantitative description of wavemaker experiments.

Fractional Leibniz rules are reminiscent of the product rule learned in calculus classes, offering estimates in the Lebesgue norm for fractional derivatives of a product of functions in terms of the Lebesgue norms of each function and its fractional derivative. We prove such estimates for Coifman-Meyer multiplier operators in the setting of Triebel-Lizorkin and Besov spaces based on quasi-Banach function spaces. As corollaries, we obtain results in weighted mixed Lebesgue spaces and Morrey spaces, where we present applications to the specific case of power weights. Other examples also include the class of rearrangement invariant quasi-Banach function spaces, of which weighted Lebesgue spaces, Lorentz spaces, and Orlicz spaces are specific examples.

This talk focuses on the well-posedness of the derivative nonlinear Schrödinger equation on the line. This model is known to be completely integrable and $L^2$-critical with respect to scaling. However, until recently not much was known regarding the well-posendess of the equation below $H^{\frac 1 2}$. In this talk we prove that the problem is well-posed in the critical space $L^2$ on the line, highlighting several recent results that led to this resolution. This is joint work with Benjamin Harrop-Griffiths, Rowan Killip, and Monica Visan.

In 1978, S.T. Yau solved the Calabi conjecture. In particular, he proved the existence of a Ricci-flat Kähler metric, which is called Calabi-Yau metric now, in each Kähler class on a closed Kähler manifold with vanishing first Chern class. Later in 1985, H.D. Cao showed that the Kähler-Ricci flow on the same Kähler manifold has long-time existence and converges to the Calabi-Yau metric. Now the focus has been turned to noncompact manifolds. In 2014, Haskins-Hein-Nordstrom proved the existence of ACyl Calabi-Yau metrics on certain noncompact manifolds. This inspires us to investigate the Kähler-Ricci flow on ACyl manifolds, hoping it will converge to an ACyl Calabi-Yau metric and further can be used to classify ACyl manifolds. However, we first need to show the short-time existence of a flow preserving the ACyl condition. This talk will be mainly expository. I will talk a little about my current research.

Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data. In 1990's, Bressan and etc proved that the BV solution verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves, is unique. In this talk, I will first discuss the recent works with Krupa and Vasseur for systems with two unknowns and non-isentropic Euler equations with three unknowns, where we established an L2 stability theory using the method of relative entropy. As an application, we proved all BV solutions must statisfy the Bounded Variation Condition, hence showed the uniqueness of BV solution without any additional condition. Then I will briefly introduce the recent progress on the vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations. This is a famous open problem after Bressan-Bianchini’s seminal vanishing artificial viscosity limit result for BV solutions of hyperbolic conservation laws. This is a join work with Kang and Vasseur.