Global Well-posedness Research Articles

On bilinear Strichartz estimates on waveguides with applications

We study local-in-time and global-in-time bilinear Strichartz estimates for the Schrödinger equation on waveguides. As applications, we apply those estimates to study global well-posedness of nonlinear Schrödinger equations on these waveguides.

Multi-index upper semicontinuity of pullback random attractors for fractional discrete FitzHugh-Nagumo systems with delays driven by Wong-Zakai approximations

In this paper, we consider the global well-posedness and multi-index stability of pullback random attractors for random fractional retarded lattice FitzHugh-Nagumo systems with nonlinear Wong-Zakai noise and non-autonomous forcing term. We use the idea of Caraballo et al. (2014) [2] to prove the global well-posedness. Based on the global well-posedness of the lattice FitzHugh-Nagumo system, we study the existence, uniqueness and upper semicontinuity of pullback random attractors Aϱδ={Aϱδ(τ,ω):τ∈R,ω∈Ω}. More precisely, we first utilize the method of tail-estimates of solution and Ascoli-Arzelà theorem to prove the existence and uniqueness of pullback random attractors, and then investigate the four types of upper semicontinuity of pullback random attractors: (1) The long time stability of pullback random attractors as the time parameter τ approaches negative infinity; (2) The upper semicontinuity of pullback random attractors as the step-length δ of the Wong-Zakai approximation tends to positive infinity; (3) The upper semicontinuity of pullback random attractors as the delay time ϱ→0; (4) The upper semicontinuity of pullback random attractors from non-autonomous to autonomous.

Global well-posedness for a system of quasilinear wave equations on a product space

Global well-posedness for a system of quasilinear wave equations on a product space

Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.

Global Well-Posedness and Exponential Decay of Strong Solution to the Three-Dimensional Nonhomogeneous Bénard System with Density-Dependent Viscosity and Vacuum

Global Well-Posedness and Exponential Decay of Strong Solution to the Three-Dimensional Nonhomogeneous Bénard System with Density-Dependent Viscosity and Vacuum

Global well-posedness of three-dimensional magneto-micropolar fluid equations with partial dissipation

In this paper, the three-dimensional magneto-micropolar fluid equations with partial dissipation is studied on R 3 . The main purpose of this paper is to establish global solutions with a small initial value. More specifically, we demonstrate the global stability of the perturbation solutions near the background magnetic field. The proof is mainly based on the energy estimate, the tricky interpolation techniques and bootstrapping argument.

Global Well-posedness for the Fourth-order Defocusing Cubic Equation with Initial Data Lying in a Critical Sobolev Space

Global Well-posedness for the Fourth-order Defocusing Cubic Equation with Initial Data Lying in a Critical Sobolev Space

A systematization of global well-posedness in vector optimization

A systematization of global well-posedness in vector optimization

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter α∈(0,1). This highlights the distinct feature of the alignment operator.

Global well-posedness for the derivative nonlinear Schrödinger equation in $L^{2}(\R)$

We prove that the derivative nonlinear Schrödinger equation in one space dimension is globally well-posed on the line in L^{2}(\mathbb{R}) , which is the scaling-critical space for this equation.

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity.

Global well-posedness and decay to 3D MHD equations with nonlinear damping

In this paper, we consider the Cauchy problem for the 3D MHD system with nonlinear damping terms κ|u|α−1u and γ|b|β−1b and prove the existence and uniqueness of global strong solution when one of the following two conditions is verified (1)3≤α<4,β≥6α−1+1;(2)α≥4,β≥1. This greatly improves the previous results. Moreover, we also discuss the uniqueness of weak solutions when α and β satisfy certain conditions. Finally, we study the decay of weak solutions when α,β>73 and obtain the upper bound for the derivative of strong solution when 3≤α<4,β≥6α−1+1 or α≥4,β>73.

Global well-posedness of the 3D damped micropolar Bénard system with horizontal dissipation

In this paper, we consider the global well-posedness of the 3D damped micropolar Bénard system with horizontal dissipation. Global existence and uniqueness of the solution of system (1.1) are proved for β≥4 and α>0.

Local and global well-posedness for fractional porous medium equation in critical Fourier-Besov spaces

In this paper, we study the Cauchy problem for the Fractional Porous Medium Equation in Rn for n ≥ 2. By using the contraction mapping method, Littlewood-Paley theory and Fourier analysis, we get, when 1 β ≤ 2, the local solution v ∈ XT := LT ∞(FBp,r (2 − 2m −β + n/p' )(Rn))∩ LTρ1(FBp,r s1(Rn))∩ LTρ2(FBp,r s2 ( Rn)) with 1 ≤ p &lt; ∞, 1 ≤ r ≤ ∞, and the solution becomes global when the initial data is small in critical Fourier-Besov spaces FBp,r (2 − 2m −β + n/p' )(Rn) . In addition, We establish a blowup criterion for the solutions. Furthermore, the global existence of solutions with small initial data in FB∞,1 (2 − 2m −β + n )(Rn) is also established. In the limit case β = 1, we prove global well-posedness for small initial data in critical Fourier-Besov spaces FBp,1 (2 − 2m + n/p' )(Rn) with 1 ≤ p &lt; ∞ and FB∞,1 (2 − 2m + n )(Rn), respectively.

Joint evolution of a Lorentz-covariant massless scalar field and its point-charge source in one space dimension

In this paper we prove that the static solution of the Cauchy problem for a massless real scalar field that is sourced by a point charge in 1 + 1 dimensions is asymptotically stable under perturbation by compactly-supported radiation. This behavior is due to the process of back-reaction. Taking the approach of Kiessling, we rigorously derive the expression for the force on the particle from the principle of total energy-momentum conservation. We provide a simple, closed form for the particle’s self-action, and show that it is restorative in this model, i.e. proportional to negative velocity, and causes the charge to return to rest after the radiation passes through. We establish these results by studying the joint evolution problem for the particle-scalar field system, and proving its global well-posedness and the claimed asymptotic behavior.

Global well-posedness and large time behavior for the 3D magneto-micropolar equations

This paper focuses on the global well-posedness and optimal decay estimates to the 3D magneto-micropolar equations in Besov spaces. We obtain the global existence and uniqueness of solutions in Besov spaces. Moreover, the decay estimates of the global solutions are established provided that the initial data belongs to the negative Besov space.

Decay and Global Well-Posedness of the Free-Boundary Incompressible Euler Equations with Damping

Decay and Global Well-Posedness of the Free-Boundary Incompressible Euler Equations with Damping

Global well-posedness for second grade fluids

Global well-posedness for second grade fluids

On the global well-posedness of the Calogero–Sutherland derivative nonlinear Schrödinger equation

On the global well-posedness of the Calogero–Sutherland derivative nonlinear Schrödinger equation

Global Well-Posedness and Convergence Results to a 3D Regularized Boussinesq System in Sobolev Spaces

We consider a regularized periodic three-dimensional Boussinesq system. For a mean free initial temperature, we use the coupling between the velocity and temperature to close the energy estimates independently of time. This allows proving the existence of a global in time unique weak solution. Also, we establish that this solution depends continuously on the initial data. Moreover, we prove that this solution converges to a Leray-Hopf weak solution of the three-dimensional Boussinesq system as the regularizing parameter vanishes.