We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgue spaces of certain bilinear multiplier operators and singular integrals.

This paper sheds new light on the linear instability of periodic traveling wave associated with some general one-dimensional dispersive models. By using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles with mean zero is obtained. Applications of this approach are concerning with the linear instability of cnoidal wave solutions for the modified Benjamin-Bona-Mahony and the modified Korteweg-de Vries equations. The arguments presented in this investigation has prospects for the study of the instability of periodic traveling wave of other nonlinear evolution equations.

In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schrödinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow up on a straight line and on a sphere.

In this paper we define new curvature functions on $\mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even orders. Existence result for antipodally symmetric prescribed curvature functions on $\mathbb{S}^n$ is obtained. As a corollary, the existence of a conformal metric for an antipodally symmetric prescribed $Q-$curvature functions on $\mathbb{S}^3$ is proved. Curvature functions on general compact manifolds as well as the conformal covariance property for the corresponding integral operator are also addressed.

Smooth non-autonomous $T$-periodic differential equations $x'(t)=f(t,x(t))$ defined in $ \mathbb R \times \mathbb K ^n$, where $ \mathbb K $ is $ \mathbb R $ or $\mathbb C$ and $n\ge 2$ can have periodic solutions with any arbitrary period~$S$. We show that this is not the case when $n=1.$ We prove that in the real $\mathcal{C}^1$-setting the period of a non-constant periodic solution of the scalar differential equation is a divisor of the period of the equation, that is $T/S\in \mathbb N .$ Moreover, we characterize the structure of the set of the periods of all the periodic solutions of a given equation. We also prove similar results in the one-dimensional holomorphic setting. In this situation the period of any non-constant periodic solution is commensurable with the period of the equation, that is $T/S\in \mathbb Q .$

This paper considers the analysis of the BF model of the spreading of a charged microdroplet on a flat dielectric surface whose spreading is driven by surface tension and electrostatic repulsion. This model assumes the droplets are circular and spread according to a power law. This leads to a third order nonlinear ordinary differential equation on $[0,1]$ that gives the evolution of the height profile. We prove that there is only one such solution that is smooth on all of $[0,1]$. There are other solutions that are continuous on $[0,1]$, but not differentiable at $x=1$. For these we describe the precise behavior of the solutions at $x=1$.

We study the $\omega$-limit set of solutions of a nonlocal ordinary differential equation, where the nonlocal term is such that the space integral of the solution is conserved in time. Using the monotone rearrangement theory, we show that the rearranged equation in one space dimension is the same as the original equation in higher space dimensions. In many cases, this property allows us to characterize the $\omega$-limit set for the nonlocal differential equation. More precisely, we prove that the $\omega$-limit set only contains one element.

In this paper, we study the classical three-dimensional (3D) incompressible Navier-Stokes equations. The main goal of the present paper is to establish the regularity criterion to the 3D Navier-Stokes equations via one velocity component.

We consider the 3D incompressible Hall-MHD system and prove a stability theorem for global large solutions under a suitable integrable hypothesis in which one of the parcels is linked to the Hall term. As a byproduct, a class of global strong solutions is obtained with large velocities and small initial magnetic fields. Moreover, we prove the local-in-time well-posedness of $H^{2} $-strong solutions which improves previous regularity conditions on initial data.