We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussingcase, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.

We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127-1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091-2102]. As an application, we show a local smoothing estimate for a solution of the linear Schr\"odinger equation under the assumption that the initial datum has additional angular regularity.

We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\rm bmo$-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space involving an unbounded and non-decaying vorticity. Next, we establish an estimate with a logarithmic loss of regularity for the transport equation in a bmo-type space by developing classical analysis tool such as the John-Nirenberg inequality. We also optimize estimates of solutions to the vorticity-stream formulation of the two-dimensional Euler equations with a bi-Lipschitz vector field in bmo-type space by combining an observation introduced in \cite{Y1} by Yodovich with the so-called ``quasi-conformal property" of the incompressible flow.

In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type L\'evy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly.

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion (−Δ)^α. First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with 5/6 <α ≤1for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this so-lution is smooth in R^3×(0, +∞). In particular, when α =1, we prove that the solution constructed by Korobkov and Tsai (2016) [16]satisfies the decay estimate by establishing regu-larity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in Jia and Šverák (2014) [13].

We study the $L^p$-theory for the Schr\"odinger operator with critical rough potential of $a|x|^{-2}$ type. The developed harmonic analysis tools, such as multiplier estimate, Littlewood-Paley theory and the equivalence of Sobolev norms, will be employed to study the scattering theory of energy-critical defocusing nonlinear Schr\"odinger equation with the inverse-square potential. These tools in $L^p$-theory are new and the range of $p$ depends on the parameter $a$, which is different from the classical theory. The main difficulty is raised from the failure of Gaussian boundedness of the heat kernel associated with the operator $P_a=-\Delta+a|x|^{-2}$ when $a$ is negative.

In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schrödinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767–808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 265–298, 2015) to four dimension without radial assumption and the proof of scattering is based on the interactionMorawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions

In this paper, we continue the study of the dynamics of the traveling waves for nonlinear Schrödinger equation with derivative (DNLS) in the energy space. Under some technical assumptions on the speed of each traveling wave, the stability of the sum of two traveling waves for DNLS is obtained in the energy space by Martel–Merle–Tsai’s analytic approach in Martel et al. (Commun Math Phys 231(2):347–373, 2002, Duke Math J 133(3):405–466, 2006). As a by-product, we also give an alternative proof of the stability of the single travelingwave in the energy space in Colin andOhta (Ann InstHenri Poincaré Anal Non Linéaire 23(5):753–764, 2006), where Colin and Ohta made use of the concentration compactness argument.

We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

We obtain an improvement of the bilinear estimates of Burq, Gérard and Tzvetkov[6]in the spirit of the refined Kakeya–Nikodym estimates [2]of Blair and the second author. We do this by using microlocal techniques and a bilinear version of Hörmander’s oscillatory integral theorem in [7].