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We prove global existence and a modified scattering property for the solutions of the 2D gravity water waves system in the infinite depth setting for a class of initial data, which is only required to be small above the level \dot{H}^{1/5}\times \dot{H}^{1/2+1/5}. No assumption is made below this level. Therefore, the nonlinear solution can have infinite energy. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu‐Pusateri, Alazard‐Delort, and Ifrim‐Tataru is removed.

This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term -ν(-Δ) α/2 ρ (1 < α < 2). Firstly, the global existence of weak solutions is proved for the initial density ρ0 ∈ L01∩L d/α (60d) (d ≥ 2) with [norm of matrix]ρ0[norm of matrix] d/α < K, where K is a universal constant only depending on d, α, ν. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in Lr for any 1 < r < ∞. Secondly, for the more general initial data ρ0 ∈ L01 ∩ L05(60d) (d = 2, 3), the local existence is obtained. Thirdly, for ρ0 ∈ L01 (60d; (1 + |x|)dx ∩ L∞(60d)( d ≥ 2) with [norm of matrix]ρ0[norm of matrix]d/α < K, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant α-stable Lévy process Lα(t). Also, we prove the weak solution is L1 bounded uniformly in time. Lastly, we consider the N-particle interacting system with the Lévy process Lα(t) and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment ∫60d |x| γρ0dx for some 1 < γ < α is below a universal constant K γ and ν is also below a universal constant. Meanwhile, we prove the propagation of chaos as N → ∞ for the interacting particle system with a cut-off parameter ε ~ (ln N)-1/d, and show that the mean field limit equation is exactly the generalized KS equation.

Abstract We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $$ \int_\Omega u^{2\gamma-\alpha-\beta}\Delta\ua\Delta\ub dx \geq c\int_\Omega|\Delta \ur |^2dx, $$ which seem to be of interest on their own right.

The uniqueness of positive solutions to some semilinear elliptic systems with variational structure arising from mathematical physics is proved. The key ingredient of the proof is the oscillatory behavior of solutions to linearized equations for cooperative semilinear elliptic systems of two equations on one-dimensional domains, and it is shown that the stability of the positive solutions for such semilinear system is closely related to the oscillatory behavior.