In this paper we give an estimate for the solution to the Schrödinger equation with sub-quadratic potentials in modulation spaces by the norm of the initial functions in Wiener-Amalgum spaces.
The aim of this paper is to discuss some well known explicit examples of the three dimensional inviscid flows with free boundary constructed by John \cite{John} and Ovsyannikov \cite{Ovsyannikov}, and provide a detailed analysis of its long time behaviour.
In this paper, we study the cubic defocusing nonlinear wave equation on the three dimensional hyperbolic space. We use the Fourier truncation method to show that the equation is globally well-posed and scatters if the initial data lies in $H^s(\mathbb{H}^3)$, $s>\frac{182}{201}\approx 0.905$.
We prove existence and uniqueness of solutions in Morrey spaces of functions with mixed norms for second-oder parabolic equations in the whole space with VMO $a$ and Morrey $b,c$.
The recent work [11] developed a general framework to show hypocoercivity for a stationary Gibbs state and allowed spatial degeneracy, confining potentials and boundary conditions. In this work, we show that the explicit energy approach in the weighted L$^2$ space works for general non-equilibrium steady states and that it can be adapted to cases with weaker confinement leading to algebraic decay.
The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for nonnegative, bounded measurable initial functions with compact support.
We determine the optimal constant in the $L^{2}$ Folland-Stein inequality on the H-type group, which partially confirms the conjecture given by Garofalo and Vassilev (Duke Math. J., 2001). The proof is inspired by the work of Frank and Lieb (Ann. of Math., 2012) and Hang and Wang.
In this paper, we establish the occurrence of blow-up in the time-fractional term at the quenching point. By demonstrating that the quenching points are contained within a compact subset of the designated spatial interval, we employ this finding to prove the blow-up of the Caputo fractional time-derivative at the quenching point.
We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the $3D$ incompressible Navier-Stokes equations with nonlinear exponential damping term $a (e^{b |u|^{\bf 4}}-1)u$, ($a,b>0$).
We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.
Schauder's fixed point theorem is used to derive the existence of solutions to a semilinear heat equation. The equation features a nonlinear term that depends on the time-integral of the unknown on the whole, a priori given, interval of existence.
We investigate the non-existence and existence of positive solutions to biharmonic elliptic inequalities on manifolds. Using Green function and volume growth conditions, we establish the critical exponent for biharmonic problem.
In this paper, we investigate a coupled Patlak-Keller-Segel-Navier-Stokes (PKS-NS) system. We show that globally regular solutions with arbitrary large cell populations exist. The primary blowup suppression mechanism is the shear flow mixing induced enhanced dissipation phenomena.
In this paper, we study the existence and concentration of solitary waves for a class of generalized Kadomtsev-Petviashvili equations with the potential in $\mathbb{R}^2$ via the variational methods.
We live in the information age, and our lives are increasingly digitized. Our quotidian has been transformed over the last fifty years by the adoption of innovative networking and computing technology. The digital world presents opportunities for public archaeology to engage, inform and interact with people globally. Yet, as more personal data are published online, there are growing concerns over privacy, security, and the long-term implications of sharing digital information. These concerns extend beyond the living, to the dead, and are thus important considerations for archaeologists who share the stories of past people online. This analysis argues that the ‘born-digital’ records of humanity may be considered as public digital mortuary landscapes, representing death, memorialization and commemoration. The potential for the analysis of digital data from these spaces could result in a phenomenon approaching immortality, whereby artificial intelligence is applied to the data of the dead. This paper investigates the ethics of a digital public archaeology of the dead while considering the future of our digital lives as mnemonic spaces, and their implications for the living.
We characterize generalized Young measures, the so-called DiPerna-Majda measures which are generated by sequences of gradients. In particular, we precisely describe these measures at the boundary of the domain in the case of the compactification of $\R^{m\times n}$ by the sphere.
We study cone differential operators on the half-axis and edge-degenerate differential operators on a half-space. We construct subspaces of edge Sobolev spaces that can be considered as natural domains for edge-degenerate operators and indicate how they can be used in the study of boundary problems for edge-degenerate operators.