scholarly journals Long time well-posedness of Prandtl system with small and analytic initial data

2016 ◽  
Vol 270 (7) ◽  
pp. 2591-2615 ◽  
Author(s):  
Ping Zhang ◽  
Zhifei Zhang
Keyword(s):  
Initial Data ◽  
Well Posedness ◽  
Long Time ◽  
Prandtl System

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in .H s × .H s-1, s> 1/2

2018 ◽  
Vol 2019 (21) ◽  
pp. 6797-6817
Author(s):  
Benjamin Dodson
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Three Dimensions ◽  
Well Posedness ◽  
Critical Space ◽  
Initial Value ◽  
Long Time ◽  
Well Posed

Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$ for some $s> \frac{1}{2}$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11].

scholarly journals Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces

2020 ◽  
Vol 70 (4) ◽  
pp. 877-892
Author(s):  
Jamel Benameur ◽  
Lotfi Jlali
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Well Posedness ◽  
Time Decay ◽  
Navier Stokes Equation ◽  
Gevrey Spaces ◽  
Gevrey Space ◽  
Long Time ◽  
Incompressible Navier Stokes Equation

AbstractIn this paper, we prove a global well-posedness of the three-dimensional incompressible Navier-Stokes equation under initial data, which belongs to the Lei-Lin-Gevrey space $\begin{array}{} Z^{-1}_{a,\sigma} \end{array}$(ℝ3) and if the norm of the initial data in the Lei-Lin space 𝓧−1 is controlled by the viscosity. Moreover, we will show that the norm of this global solution in the Lei-Lin-Gevrey space decays to zero as time approaches to infinity.

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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.

scholarly journals CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

2000 ◽  
Vol 09 (01) ◽  
pp. 13-34 ◽  
Author(s):  
GEN YONEDA ◽  
HISA-AKI SHINKAI
Keyword(s):  
General Relativity ◽  
Hyperbolic System ◽  
Hyperbolic Systems ◽  
Equations Of Motion ◽  
Well Posedness ◽  
Symmetric Hyperbolic System ◽  
The Cauchy Problem ◽  
Long Time ◽  
Vacuum Spacetime ◽  
Gauge Conditions

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

scholarly journals Well posedness for the Kawahara equation on the half-line

2020 ◽  
Author(s):  
Boling Guo ◽  
fengxia liu
Keyword(s):  
Initial Data ◽  
Local Existence ◽  
Half Line ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Low Regularity ◽  
Regularity Properties

We study the low-regularity properties of the Kawahara equation on the half line. We obtain the local existence, uniqueness, and continuity of the solution. Moreover, We obtain that the nonlinear terms of the solution are smoother than the initial data.

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