Small Initial Data Research Articles

On the lifespan of nonzero background solutions to a class of focusing nonlinear Schrödinger equations

The global solvability in time and the potential for blow-up of solutions to non-integrable focusing nonlinear Schrödinger equations with nonzero boundary conditions at infinity present challenges that are less explored and understood compared to the case of zero boundary conditions. In this work, we address these questions by establishing estimates on the lifespan of solutions to non-integrable equations involving a general class of nonlinearities. These estimates depend on the size of the initial data, the growth of the nonlinearity, and relevant quantities associated with the amplitude of the background. The estimates provide quantified upper bounds for the minimum guaranteed lifespan of solutions. Qualitatively, for small initial data and background, these upper bounds suggest long survival times consistent with global existence of solutions. On the other hand, for larger initial data and background, the estimates indicate the potential for the intriguing phenomenon of instantaneous collapse in finite time. These qualitative theoretical results are illustrated via numerical simulations. Furthermore, importantly, the numerical findings motivate the proof of improved theoretical upper bounds that provide excellent quantitative agreement with the order of the numerically identified lifespan of solutions.

Nonlinear Memory Term for Fractional Diffusion Equation

This paper examines the Cauchy problem described by the following equation: ∂tλ+1ϕ-Δϕ=∫0t(t-s)-γϕ(s,.)pds,ϕ(0,x)=ϕ0(x),ϕt(0,x)=ϕ1(x).(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _{t}^{\\lambda +1}\\phi -\\Delta \\phi =\\int _{0}^{t}(t-s)^{- \\gamma } \\left| \\phi (s,.) \\right| ^{p}ds,\\quad \\phi (0,x)=\\phi _{0}(x),\\quad \\phi _{t}(0,x)=\\phi _{1}(x). \\quad \\mathrm{(1)} \\end{aligned}$$\\end{document}The equation involves the Caputo fractional derivative in time, denoted as ∂tλ+1ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _{t}^{\\lambda +1}\\phi$$\\end{document}. Additionally, The nonlinear term is determined by the memory term ∫0t(t-s)-γϕ(s,.)pds\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{0}^{t}(t-s)^{- \\gamma } \\left| \\phi (s,.) \\right| ^{p}ds$$\\end{document}, where γ∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\in (0,1)$$\\end{document}. Using the fixed point theorem, we establish the global existence of solutions to the Cauchy problem (1) for small initial data. We also investigate the impact of the nonlinearity parameter on the range of the exponent p and the estimation of the solutions.

Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation

In this paper, we consider the 1D Euler equation with time and space dependent damping term −a(t,x)v. It has long been known that when a(t,x) is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient a satisfies the following condition |a(t,x)−μ0|≤a1(t)+a2(x),where μ0≥0 and a1 and a2 are integrable functions with t and x. Under this condition, we show the global existence and the blow-up with small initial data, when μ0>0 and μ0=0 respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.

Global well-posedness for the three dimensional compressible micropolar equations

In this paper, we study the Cauchy problem of the three-dimensional compressible micropolar equations in the absence of heat-conductivity. By leveraging Fourier theory and employing a refined energy method, we establish the global well-posedness of the equations for small initial data within Besov spaces. As a byproduct, we also derive the optimal time decay of solutions if the low frequency of initial data belonging to Ḃ2,∞−σ1(R3).

Global solutions to the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner system with a class of large initial data

In this paper, we consider the global well-posedness of the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner (PTT) system with general initial data in the critical Besov spaces. The question of whether or not PTT system is globally well posed for large data is still open. When the density ρ is away from zero, we denote by ϱ:=1ρ−1. More precisely, we prove that the PTT system admits a unique global solution, provided that the initial data ϱ 0, the initial horizontal velocity uh0 , the product ωu30 of the coupling parameter ω and the initial vertical velocity u 30, and the initial symmetric tensor of constraints τ 0 are sufficiently small. In particular, this result includes the global well-posedness of the PTT system for small initial data in the case where ω∈[0,1) and for large initial vertical velocity in the case where ω tends to zero. As a by-product, our results can be applied to the so-called Oldroyd-B system.

Well-posedness result for the flow of Maxwell viscoelastic fluid of White-Metzner type

In this paper, we investigate the flow characteristics of an unsteady quasistatic Maxwell viscoelastic fluid of White-Metzner type within a three-dimensional bounded domain. Our objective is to establish the well-posedness of the resulting partial differential equations initial boundary value problem, both locally and globally in time, under the condition of small initial data. To demonstrate the existence of a solution, we employ an iterative method that transforms our problem into a sequence of linear subproblems. At each iteration, we tackle both elliptic and hyperbolic problems. The convergence of this iterative process provides the solution to the original problem.

Global solution for semilinear σ$$ \sigma $$‐evolution models with memory term

In this paper, the initial value problem of the semilinear ‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data.

Local and global existence for the Ericksen - Leslie problem in unbounded domains

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the space RN with N≥3, on R+3 and on Ω⊆R3 exterior domain with sufficiently smooth boundary. The crystal orientation is described by unit vector v that is a small perturbation of a fixed constant vector η. We prove through a combination of energy method with dispersive a priori estimates a local existence and global existence for small initial data by a contraction argument. In particular, we obtain the following regularity of the liquid velocity u and of the crystal orientation vu∈L∞((0,T);Hs(Ω)),∇u∈L2((0,T);Hs(Ω)),∇v∈L∞((0,T);Hs(Ω)),∇2v∈L2((0,T);Hs(Ω)) for s>N2−1 if Ω=RN and s∈(12,1] if Ω=R+3 or in the exterior case, asking low regularity assumptions on u0 and v0.

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 existence of nonlinear elastic waves: Non-hyperelastic case

For the Cauchy problem of nonlinear elastic wave equations for three-dimensional isotropic, homogeneous and hyperelastic materials with null condition, global existence of classical solutions with small initial data was proved in Agemi (2000) [1] and Sideris (2000) [17] independently. How to treat the more general non-hyperelastic case is suggested as an open problem in Agemi (2005) [2]. The aim of this paper is to resolve this problem.

Non‐global solutions for the inhomogeneous nonlinear Schrödinger equation without gauge invariance

This work investigates the non‐global existence of solutions for the inhomogeneous nonlinear Schrödinger problem without gauge invariance. Indeed, initially, we establish the non‐global existence of certain mass‐subcritical solutions. Subsequently, we demonstrate the non‐global existence of energy‐subcritical solutions characterized by small initial data. These results are are acknowledged to be unsuccessful for the classical gauge invariant inhomogeneous nonlinear Schrödinger equation (INLS).

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 < ∞, 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 < ∞ and FB∞,1 (2 − 2m + n )(Rn), respectively.

Exact global control of small divisors in rational normal form *

Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schrödinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.

Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations *

This paper is devoted to some inverse problems of recovering the nonlinearity for the Jordan–Moore–Gibson–Thompson equation, which is a third order nonlinear acoustic equation. This equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained with the small initial and boundary data. By the second order linearization to the nonlinear equation, and construction of complex geometric optics solutions for the linearized equation, the uniqueness of recovering the nonlinearity is derived.

Stability and exponential decay for the compressible viscous non-resistive MHD system

How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in R3 or T3 is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in T3 when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.

On the Mass Transfer in the 3D Pitaevskii Model

We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in T3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}^3$$\\end{document} for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

Global well-posedness for the inhomogeneous incompressible MHD equations with variable viscosity coefficient and electrical conductivity

In this paper, we study the Cauchy problem for the inhomogeneous incompressible MHD system with variable viscosity coefficient and electrical conductivity. It is shown that the 3-D inhomogeneous incompressible MHD system has unique global solutions with small initial data conditions. Furthermore, we establish the global well-posedness of the two-dimensional MHD equations in critical spaces, given certain conditions on the density.

Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle

AbstractWe consider a class of Hamiltonian Klein–Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small initial data of size $$\varepsilon $$ ε showing that the corresponding solutions remain close to oscillatory motions over a time scale $$\varepsilon ^{{-\frac{9}{4}+\delta }}$$ ε - 9 4 + δ for any $$\delta >0$$ δ > 0 . The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach.

Long time stability for the derivative nonlinear Schrödinger equation

In this paper, we consider the long time dynamics of the solutions of the derivative nonlinear Schrödinger equation on one dimensional torus without external parameters. By using rational normal form, we prove the long time stability for generic small initial data.

Diffusive stability and self-similar decay for the harmonic map heat flow

In this paper we study the harmonic map heat flow on the euclidean space Rd and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B˙p,∞dp(Rd) where d<p<∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖∇u(t)‖L∞(Rd)≤Ct−12.Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.