scholarly journals Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS

2006 ◽  
Vol 5 (4) ◽  
pp. 691-708 ◽  
Author(s):  
P. Blue ◽  
◽  
J. Colliander
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Global Existence ◽  
Well Posedness

Local well-posedness for the quantum Zakharov system in one spatial dimension

2017 ◽  
Vol 14 (01) ◽  
pp. 157-192 ◽  
Author(s):  
Yung-Fu Fang ◽  
Hsi-Wei Shih ◽  
Kuan-Hsiang Wang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Electric Field ◽  
Classical Result ◽  
Spatial Dimension ◽  
Well Posedness ◽  
Ion Density ◽  
Zakharov System ◽  
Quantum Parameter ◽  
Quantum Zakharov System

We consider the quantum Zakharov system in one spatial dimension and establish a local well-posedness theory when the initial data of the electric field and the deviation of the ion density lie in a Sobolev space with suitable regularity. As the quantum parameter approaches zero, we formally recover a classical result by Ginibre, Tsutsumi, and Velo. We also improve their result concerning the Zakharov system and a result by Jiang, Lin, and Shao concerning the quantum Zakharov system.

Global well-posedness of the 2-D magnetic Prandtl model in the Prandtl–Hartmann regime

2020 ◽  
Vol 120 (3-4) ◽  
pp. 373-393
Author(s):  
Dongxiang Chen ◽  
Siqi Ren ◽  
Yuxi Wang ◽  
Zhifei Zhang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Small Perturbation ◽  
Well Posedness ◽  
Prandtl Model

In this paper, we prove the global well-posedness of the 2-D magnetic Prandtl model in the mixed Prandtl/Hartmann regime when the initial data is a small perturbation of the Hartmann layer in Sobolev space.

Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4

2020 ◽  
Vol 32 (6) ◽  
pp. 1575-1598
Author(s):  
Zhaohui Huo ◽  
Yueling Jia
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Well Posedness ◽  
Small Initial Data ◽  
Bilinear Estimates ◽  
The Cauchy Problem ◽  
Univariate Polynomial ◽  
Well Posed ◽  
Resonance Function

AbstractThe Cauchy problem of the 2D Zakharov–Kuznetsov equation {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space {H^{-1/4}}, and it is globally well-posed in {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

scholarly journals Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Yongqiang Xu
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Well Posedness ◽  
Two Dimensional ◽  
Small Initial Data ◽  
General Initial Data ◽  
Local Existence Of Solutions

This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when1/2≤α<1.

A global existence of classical solutions to the hydrodynamic Cucker–Smale model in presence of a temperature field

2018 ◽  
Vol 16 (06) ◽  
pp. 757-805 ◽  
Author(s):  
Seung-Yeal Ha ◽  
Jeongho Kim ◽  
Chanho Min ◽  
Tommaso Ruggeri ◽  
Xiongtao Zhang
Keyword(s):  
Temperature Field ◽  
Sobolev Space ◽  
Global Existence ◽  
Hydrodynamic Model ◽  
Energy Estimates ◽  
Classical Solutions ◽  
Weight Functions ◽  
Time Interval ◽  
Well Posedness ◽  
Non Local

We present a hydrodynamic model for the ensemble of thermodynamic Cucker–Smale (TCS) particles in the presence of a temperature field, and study its global-in-time well-posedness in Sobolev space. Our hydrodynamic model can be formally derived from the kinetic TCS model under the mono-kinetic ansatz, and can be viewed as a pressureless gas dynamics with non-local flocking forces. For the global-in-time well-posedness, we assume that communication weight functions are non-negative and non-increasing in their arguments and initial data satisfy non-vacuum conditions and suitable regularity in Sobolev space. In this setting, we use the method of energy estimates and obtain the global existence of classical solutions in any finite time interval. We also present an asymptotic flocking estimate using the Lyapunov functional approach.

scholarly journals Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces

2017 ◽  
Vol 8 (1) ◽  
pp. 203-224 ◽  
Author(s):  
Yuzhao Wang ◽  
Jie Xiao
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Carleson Measure ◽  
Stokes System ◽  
Mild Solutions ◽  
Navier Stokes ◽  
Well Posedness ◽  
Solution Map ◽  
Measure Spaces ◽  
Dissipative Case

Abstract As an essential extension of the well known case {\beta\kern-1.0pt\in\kern-1.0pt({\frac{1}{2}},1]} to the hyper-dissipative case {\beta\kern-1.0pt\in\kern-1.0pt(1,\infty)} , this paper establishes both well-posedness and ill-posedness (not only norm inflation but also indifferentiability of the solution map) for the mild solutions of the incompressible Navier–Stokes system with dissipation {(-\Delta)^{{\frac{1}{2}}<\beta<\infty}} through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space {(Q_{-s=-\alpha})^{n}} , the BMO-Sobolev space {((-\Delta)^{-{\frac{s}{2}}}\mathrm{BMO})^{n}} , the Lip-Sobolev space {((-\Delta)^{-{\frac{s}{2}}}\mathrm{Lip}\alpha)^{n}} , and the Besov space {(\dot{B}^{s}_{\infty,\infty})^{n}} .

scholarly journals On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
David Massatt
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Two Dimensional ◽  
Periodic Domain ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
Sivashinsky Equation

<p style='text-indent:20px;'>We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data <inline-formula><tex-math id="M1">\begin{document}$ u_{01} \in L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ \eta &gt; 0 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Wave-breaking phenomena and global existence for the weakly dissipative generalized Novikov equation

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Shuguan Ji ◽  
Yonghui Zhou
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Global Solution ◽  
Wave Breaking ◽  
Blow Up ◽  
Momentum Density ◽  
Well Posedness ◽  
Novikov Equation ◽  
Blow Up Rate

In this paper, we mainly study several problems on the weakly dissipative generalized Novikov equation. We first establish the local well-posedness of solutions. We then give the precise blow-up scenarios for the generalized Novikov equation provided the momentum density associated with their initial data changes sign, and obtain the blow-up rate of blow-up solutions. Finally, we prove that the equation has a global solution provided the momentum density associated with their initial data do not change sign.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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