Small Data Global Well-posedness Research Articles

A note on the $H^{s}$-critical inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the Cauchy problem for the H^{s} -critical inhomogeneous nonlinear Schrödinger (INLS) equation iu_{t} +\Delta u=\lvert{x}\rvert^{-b}f(u),\quad u(0)=u_{0} \in H^{s} (\mathbb{R}^{n}), where n\in \mathbb{N} , 0\le s<\smash{\frac{n}{2}} , 0<b<\min\{2, n-s, 1+\smash{\frac{n-2s}{2}}\} and f(u) is a nonlinear function that behaves like \lambda \lvert{u}\rvert^{\sigma}u with \lambda \in \mathbb{C} and \sigma=\smash{\frac{4-2b}{n-2s}} . First, we establish the local well-posedness as well as the small data global well-posedness in H^{s}(\mathbb{R}^{n}) for the H^{s} -critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev–Lorentz spaces. Next, we obtain some standard continuous dependence results for the H^{s} -critical INLS equation. Our results about the well-posedness and standard continuous dependence for the H^{s} -critical INLS equation improve the ones of Aloui–Tayachi [Discrete Contin. Dyn. Syst. 41 (2021), 5409–5437] and An–Kim [Evol. Equ. Control Theory 12 (2023), 1039–1055] by extending the validity of s and b . Based on the local well-posedness in H^{1}(\mathbb{R}^{n}) , we finally establish the blow-up criteria for H^{1} -solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.

Global well-posedness of 2D incompressible Oldroyd-B model with only velocity dissipation

The small data global well-posedness in Sobolev setting for the incompressible Oldroyd-B equations with only velocity dissipation (namely there is no damping or dissipation of stress tensor) on R2 remains open by now. The approach for the similar problem on R3 becomes invalid due to the fewer decay rates. In this paper, we will give a positive answer to this problem. Namely, we shall focus on the Cauchy problem for the incompressible Oldroyd-B equations with only velocity dissipation on R2 and derive the small data global well-posedness under Sobolev setting. We also give the time decay estimate for the solutions. To achieve this goal, we explore the dissipative structure of system and develop the fractional time-weighted energy framework. Moreover, some delicate commutator and bilinear estimates are proposed to deal with the wildest nonlinear terms.

Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in $ H^{s}(\mathbb R^{n}) $

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation$ iu_{t} +\Delta u = |x|^{-b} f(u), \; u(0) = u_{0} \in H^{s} (\mathbb R^{n} ), $where $ n\ge3 $, $ 1< s<\frac{n}{2} $, $ 0<b<2 $ and $ f(u) $ is a nonlinear function that behaves like $ \lambda |u|^{\sigma } u $ with $ \lambda \in \mathbb C $ and $ \sigma = \frac{4-2b}{n-2s} $. We establish the local well-posedness as well as the small data global well-posedness and scattering in $ H^{s} (\mathbb R^{n} ) $ with $ 1<s<\frac{n}{2} $ for the critical INLS equation under some assumption on $ b $. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.

Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

<p style='text-indent:20px;'>In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS)</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta^{2} u = \lambda |x|^{-b}|u|^{\sigma}u, u(0) = u_{0} \in H^{s} (\mathbb R^{d}), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda \in \mathbb R $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d\in \mathbb N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0<s<\min \{2+\frac{d}{2}, \frac{3}{2}d\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ 0<b<\min\{4, d, \frac{3}{2}d-s, \frac{d}{2}+2-s\} $\end{document}</tex-math></inline-formula>. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in <inline-formula><tex-math id="M5">\begin{document}$ H^{s}(\mathbb R^{d}) $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M6">\begin{document}$ \frac{8-2b}{d}<\sigma< \sigma_{c}(s) $\end{document}</tex-math></inline-formula> and the initial data is sufficiently small, where <inline-formula><tex-math id="M7">\begin{document}$ \sigma_{c}(s) = \frac{8-2b}{d-2s} $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M8">\begin{document}$ s<\frac{d}{2} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ \sigma_{c}(s) = \infty $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M10">\begin{document}$ s\ge \frac{d}{2} $\end{document}</tex-math></inline-formula>.</p>

Global well-posedness and scattering for the Zakharov system at the critical space in three spatial dimensions with small and radial initial data

We consider the Cauchy problem for the Zakharov system at the critical space in three spatial dimensions. Here, the initial data are small and spherically symmetric. When k and l are the regularity of the initial data of the Schrödinger equation and the wave equation, the critical exponent of the three-dimensional Zakharov system is (k,l)=(0,−12). If the initial data are spherically symmetric, we can show the small data global well-posedness and the scattering in the space with the critical exponent by applying the radial Strichartz estimates.

Well-posedness for the Cauchy Problem of the Modified Zakharov-Kuznetsov Equation

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s \geq 1/4$. If $d \geq 3$, by employing $U^p$ and $V^p$ spaces, we establish the small data global well-posedness in the scaling critical Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c = d/2-1$.

On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior

This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\mathbb R^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\mathbb R^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution $u\equiv 0$ is shown to be asymptotically stable in $L^{q_c}(\mathbb R^{n})$ -- it is the only self-similar solution which is initially small in $L^{q_c}(\mathbb R^{n})$. Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.

Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$($0\leq s<\frac {3}{2}$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

Small data scattering of 2d Hartree type Dirac equations

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity c(|⋅|−γ⁎〈ψ,βψ〉)βψ with c∈R∖{0}, 0<γ<2. Our aim is to show the small data global well-posedness and scattering in Hs for s>γ−1 and 1<γ<2. The difficulty stems from the singularity of the low-frequency part |ξ|−(2−γ)χ{|ξ|≤1} of potential. To overcome it we adapt Up−Vp space argument and bilinear estimates of [27,25] arising from the null structure. We also provide nonexistence result for scattering in the long-range case 0<γ≤1.

On global well-posedness and decay of 3D Ericksen-Leslie system

&lt;abstract&gt;&lt;p&gt;In this paper, the small initial data global well-posedness and time decay estimates of strong solutions to the Cauchy problem of 3D incompressible liquid crystal system with general Leslie stress tensor are studied. First, assuming that $ \|u_0\|_{\dot{H}^{\frac12+\varepsilon}}+\|d_0-d_*\|_{\dot{H}^{\frac32+\varepsilon}} $ ($ \varepsilon &amp;gt; 0) $ is sufficiently small, we obtain the global well-posedness of strong solutions. Moreover, the $ L^p $–$ L^2 $ ($ \frac32\leq p\leq2 $) type optimal decay rates of the higher-order spatial derivatives of solutions are also obtained. The $ \dot{H}^{-s} $ ($ 0\leq s &amp;lt; \frac12 $) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.&lt;/p&gt;&lt;/abstract&gt;

Global well-posedness and large time decay for the d-dimensional tropical climate model

<abstract> This paper investigates the Cauchy problem on the d-dimensional tropical climate model with fractional hyperviscosity. We establish the small data global well-posedness of solutions to this model with supercritical dissipation. Furthermore, we study the asymptotic stability of these global solutions and obtain the optimal decay rates by using energy method and the method of bootstrapping argument. </abstract>

Random data theory for the cubic fourth-order nonlinear Schrödinger equation

We consider the cubic nonlinear fourth-order Schrödinger equation \begin{document}$ i \partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 $\end{document} on $ \mathbb R^N, N\geq 5 $ with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

Global well-posedness and scattering for the Dysthe equation in [formula omitted

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space L2(R2). This result is sharp in view of the fact that the flow map cannot be C3 continuous below L2(R2). Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space US2 and its dual VS2 of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in Hs(R2), s>0. Our results extend to the finite depth version of the Dysthe equation.

Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium

This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence of the techniques introduced in this paper, we obtain the small data global well-posedness for the 3D incompressible Navier-Stokes equations without vertical dissipation.

The hyperbolic Yang–Mills equation in the caloric gauge : local well-posedness and control of energy-dispersed solutions

This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy concentration scale. Moreover, we show that regularity and dispersive behavior of the solution persists as long as energy dispersion is small. We also observe that fixed-time regularity (but not dispersive) properties in the caloric gauge may be transferred to the temporal gauge without any loss, proving as a consequence small data global well-posedness in the temporal gauge. The results in this paper are used in the subsequent papers arXiv:1709.08604, arXiv:1709.08606 to prove the sharp Threshold Theorem in caloric gauge in the trivial topological class, and the dichotomy theorem in arbitrary topological classes.

Scattering results for Dirac Hartree-type equations with small initial data

We consider the Dirac equations with cubic Hartree-type nonlinearity which are derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data global well-posedness and scattering results in the full scaling subcritical regularity regime. The strategy of the proof relies on the localized Strichartz estimates and bilinear estimates in \begin{document}$ V^2 $\end{document} spaces, together with the use of the null structure that the nonlinear term exhibits. This result is shown to be almost optimal in the sense that the iteration method based on Duhamel's formula fails over the supercritical range.

On the Division Problem for the Wave Maps Equation

We consider Wave Maps into the sphere and give a new proof of small data global well-posedness and scattering in the critical Besov space, in any space dimension $n \geq 2$. We use an adapted version of the atomic space $U^2$ as the single building block for the iteration space. Our approach to the so-called division problem is modular as it systematically uses two ingredients: atomic bilinear (adjoint) Fourier restriction estimates and an algebra property of the iteration space, both of which can be adapted to other phase functions.

The derivative nonlinear Schrödinger equation on the half line

We study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on R and prove smoothing estimates by combining the restricted norm method with a normal form transformation.

Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects

In this paper, we consider the small initial data global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in $$\mathbb {R}^3$$ . In addition, we also establish the temporal decay estimates for the weak solutions. With these estimates in hand, we study the algebraic time decay for higher-order Sobolev norms of small initial data solutions.