Global Well-posedness Research Articles

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

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

Global Well-Posedness and Vanishing Normal Stress Coefficients for the Hydrostatic Second-Grade Fluid Equations

Global Well-Posedness and Vanishing Normal Stress Coefficients for the Hydrostatic Second-Grade Fluid Equations

Global Well-Posedness and Regularity of Weak Solutions to the Prandtl’s System

Global Well-Posedness and Regularity of Weak Solutions to the Prandtl’s System

Global Well-Posedness of Classical Solutions to the Compressible Navier–Stokes–Poisson Equations with Slip Boundary Conditions in 3D Bounded Domains

Global Well-Posedness of Classical Solutions to the Compressible Navier–Stokes–Poisson Equations with Slip Boundary Conditions in 3D Bounded Domains

Exponential convergence to steady-states for trajectories of a damped dynamical system modeling adhesive strings

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.

Existence and asymptotic behavior for L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term which preserves the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm of the solution. We study the local and global well-posedness on a bounded domain, as well as the whole Euclidean space, in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document} to a stationary state. For a ball, we prove strong convergence to the ground state when the initial condition is positive.

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Sharp Global Well-Posedness for the Cubic Nonlinear Schrödinger Equation with Third Order Dispersion

Sharp Global Well-Posedness for the Cubic Nonlinear Schrödinger Equation with Third Order Dispersion

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1 p>1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p > 1 p>1 and scattering for p > 3 p>3 (actually even for 1 > p ≤ 3 1>p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

Global well-posedness of three-dimensional incompressible Boussinesq system with temperature-dependent viscosity

In this paper, we focus on the global well-posedness of solutions to three-dimensional incompressible Boussinesq equations with temperature-dependent viscosity under the smallness assumption of initial velocity fields u0 in the critical space Ḃ3,10. The key ingredients here lie in the decomposition of the velocity fields and the regularity theory of the Stokes system, which are crucial to get rid of the smallness restricition of the initial temperature θ0. In addition, we mention that the improved decay estimates in time is also necessary.

Global well-posedness for the 3D inhomogeneous incompressible magnetohydrodynamics system with axisymmetric data

We focus on the Cauchy problem of 3D inhomogeneous incompressible magnetohydrodynamics (MHD for short) system with axisymmetric initial data. It is shown that this MHD system has a unique global solution provided that ‖1r(1ρ0−1)‖L∞ is sufficiently small. Finally, we establish the L2 decay rate for any given globally smooth solution of the MHD system by using the Schonbek's Fourier splitting method.

Weak Mean Random Attractor of Reversible Selkov Lattice Systems Driven By Locally Lipschitz Lévy Noises

This paper is concerned with weak pullback mean random attractor of reversible Selkov lattice systems defined on the entire integer set \(\mathbb{Z}\) driven by locally Lipschitz Lévy noises. Firstly, we formulate the stochastic lattice equations to an abstract system defined in the non-concrete space \(\ell^2\times\ell^2\) of square-summable sequences. Secondly, we establish the global well-posedness of the systems with locally Lipschitz diffusion terms. Under certain conditions, we show that the long-time dynamics can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space \(L^2(\Omega,\ell^2\times\ell^2)\). To overcome the difficulty caused by the drift and diffusion terms, we adopt a stopping time technique to prove the convergence of solutions in probability. The mean random dynamical systems theory proposed by Wang (J. Differ. Equ., 31:2177-2204, 2019) is used to deal with the difficulty caused by the nonlinear noise.

Research on p order nonlinear half wave Schrödinger equations

The objective of our research is to scrutinize the presence and singularity of the comprehensive potent solution for the half-wave Schrödinger equation characterized by  order. Take into account the nonlinear half-wave Schrödinger equations represented by  order: . We apply the Brezis-Gallouet style inequality to attain a logarithmic form of regulation. These rationales will be pivotal in validating our primary theorem within Global Well-Posedness. When discussing global Solutions, we infer the solutions for Schrödinger equations within the  realm when . Specifically, when , we leverage the Strichartz estimations alongside inequalities following the Brezis-Gallouët pattern to ascertain the well-posedness of Schrödinger equations within the  framework. To elucidate the global well-posedness of Schrödinger equations in the energy subspace , we undertake a traditional tactic to craft the frail solution within the  realm, succeeded by the application of a rationale found in existing academic literature to verify the singularity of the frail solution. The coherent progression of the frail solution is derived from the sustained conservation of both mass and energy elements. An analogous strategy is likewise harnessed to affirm the global well-posedness of Schrödinger equations in the  sphere when .

Free boundary problems of the incompressible Navier–Stokes equations with non-flat initial surface in the critical Besov space

AbstractGlobal well-posedness of the Navier–Stokes equations with a free boundary condition is considered in the scaling critical homogeneous Besov spaces $${\dot{B}}_{p,1}^{-1+n/p}({\mathbb {R}}^n_+)$$ B ˙ p , 1 - 1 + n / p ( R + n ) with $$n-1< p< 2n-1$$ n - 1 < p < 2 n - 1 . To show the global well-posedness, we establish end-point maximal $$L^1$$ L 1 -regularity for the initial-boundary value problem of the Stokes equations. Such an estimate is obtained via related estimate for the initial-boundary value problem of the heat equation with the inhomogeneous Neumann data as well as the pressure estimate in the critical Besov space framework. The proof heavily depends on the explicit expression of the fundamental integral kernel of the Lagrange transformed linearized Stokes equations and the almost orthogonal estimates with the space-time Littlewood–Paley dyadic decompositions. Our result here improves the initial space and boundary state than previous results by Danchin–Hieber–Mucha–Tolksdorf (Free boundary problems via Da Prato–Grisvard theory. arXiv:2011.07918v2) and ourselves (Ogawa and Shimizu in J Evol Equ 22(30):67, 2022; Ogawa and Shimizu in J Math Soc Jpn. arXiv:2211.06952v3).

Global well-posedness and large time behavior for the inviscid Oldroyd-B model

In this paper, we consider the 3-dimensional incompressible inviscid Oldroyd-B model. Firstly, we establish the global existence of the solutions for the inviscid Oldroyd-B model with different coupling coefficient k>0. Then, we show the connection between the solution with the parameter k that is the reciprocal of Weissenberg number. On one hand, we prove that the solutions (u,τ) depend continuously on the parameter k>0, though (u,τ) corresponds to different decays rate for different k. On the other hand, when k→0 in large time, we prove that there is a gap between the L2 norm of solution uk(t,x) to the model of parameter k>0 with the L2 norm of solution u0(t,x) to the model of parameter k=0. In a word, we prove that the larger positive k induces the faster decay rate of the solution, but k cannot go to zero, or the dissipation will vanish instantly.

Global well-posedness and L 2 decay estimate of smooth solutions for the 3-D incompressible Navier–Stokes–Allen–Cahn system

We consider the Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.

Global well-posedness for Ericksen-Leslie system with zero viscosity

We study the global well-posedness of the simplified Ericksen-Leslie system with zero viscosity on the periodic domain T 2 \mathbb {T}^2 . Precisely, we prove the global existence and uniqueness of smooth solution to Ericksen-Leslie system if the initial data ( u 0 , ∇ d 0 ) (u_0,\nabla d_0) is small in H 4 ( T 2 ) × H 4 ( T 2 ) H^4(\mathbb {T}^2)\times H^4(\mathbb {T}^2) . Furthermore, we derive the time decay estimate of ∇ d \nabla d in H 1 ( T 2 ) H^1(\mathbb {T}^2) .

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

We study a class of two-dimensional nonlinear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a nontrivial adaptation of the standard properties of Schwartz symmetrization for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.

Large time behavior for the Hall-MHD equations with horizontal dissipation

This paper examines the large time behavior of solutions to the 3D Hall-magnetohydrodynamic equations with horizontal dissipation. As preparations we establish the global well-posedness of solutions and their global explicitly uniform upper bounds for Hk (k ≥ 1) to this system with initial data small in H2. Furthermore, if the initial data also belongs to homogeneous negative Besov spaces, we prove the optimal decay rates of the aforementioned global solutions and their higher order derivatives.

Global well-posedness and interior regularity of 2D Navier–Stokes equations with stochastic boundary conditions

AbstractThe paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.