scholarly journals Global Well-Posedness for the Fractional Navier-Stokes-Coriolis Equations in Function Spaces Characterized by Semigroups

2021 ◽  
Author(s):  
Xiaochun Sun ◽  
Jia Liu ◽  
Jihong Zhang
Keyword(s):  
Mild Solutions ◽  
Function Spaces ◽  
Navier Stokes ◽  
Laplacian Operator ◽  
Well Posedness ◽  
Initial Value ◽  
General Operator ◽  
Small Initial Data ◽  
Unique Existence ◽  
Evolution Semigroup

We studies the initial value problem for the fractional Navier-Stokes-Coriolis equations, which obtained by replacing the Laplacian operator in the Navier-Stokes-Coriolis equation by the more general operator $(-\Delta)^\alpha$ with $\alpha>0$. We introduce function spaces of the Besove type characterized by the time evolution semigroup associated with the general linear Stokes-Coriolis operator. Next, we establish the unique existence of global in time mild solutions for small initial data belonging to our function spaces characterized by semigroups in both the scaling subcritical and critical settings.

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 Global Well-Posedness for Fractional Navier-Stokes Equations in critical Fourier-Besov-Morrey Spaces

2017 ◽  
Vol 3 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Azzeddine El Baraka ◽  
Mohamed Toumlilin
Keyword(s):  
Cauchy Problem ◽  
Stokes Equations ◽  
Morrey Spaces ◽  
Navier Stokes ◽  
Well Posedness ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Localization Method ◽  
Fourier Localization ◽  
The Cauchy Problem

Abstract In this paper we study the Cauchy problem of the Fractional Navier-Stokes equations in critical Fourier-Besov-Morrey spaces FṄsp, λ,q(ℝ3) with . By making use of the Fourier localization method and the Littlewood-Paley theory as in [6] and [21], we get global well-posedness result with small initial data belonging to . The space FṄsp,λ,q(ℝ3) covers the classical spaces Ḃsq and FḂsp,q(ℝ3) (cf [7],[3], [19], [22]...). The result of this paper extends the works of [6] and [21].

Asymptotic stability of small Oseen-type vortex under three-dimensional large perturbation

Analysis ◽  
2020 ◽  
Vol 40 (2) ◽  
pp. 57-83
Author(s):  
Ken Furukawa
Keyword(s):  
Asymptotic Stability ◽  
Initial Data ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Vertical Direction ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Unique Existence ◽  
Large Perturbation

AbstractWe consider the three-dimensional Navier–Stokes equations whose initial data may have infinite kinetic energy. We establish unique existence of the mild solution to the Navier–Stokes equations for small initial data in the whole space {\mathbb{R}^{3}} and a vertically periodic product space {\mathbb{R}^{2}\times\mathbb{T}^{1}} which may be constant in vertical direction so that it includes the Oseen vortex. We further discuss its asymptotic stability under arbitrarily large three-dimensional perturbation in {\mathbb{R}^{2}\times\mathbb{T}^{1}}.

ON THE WELL-POSEDNESS OF PARABOLIC EQUATIONS OF NAVIER–STOKES TYPE WITH DATA

2015 ◽  
Vol 16 (5) ◽  
pp. 947-985 ◽  
Author(s):  
Pascal Auscher ◽  
Dorothee Frey
Keyword(s):  
Parabolic Equations ◽  
Stokes System ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Well Posedness ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Tent Spaces ◽  
Quadratic Nonlinearities ◽  
Kernel Bounds

We develop a strategy making extensive use of tent spaces to study parabolic equations with quadratic nonlinearities as for the Navier–Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier–Stokes equations in $\mathbb{R}^{n}$ with small initial data in $\mathit{BMO}^{-1}(\mathbb{R}^{n})$. We then study another model where neither pointwise kernel bounds nor self-adjointness are available.

scholarly journals Well-posedness of the Navier—Stokes—Maxwell equations

2014 ◽  
Vol 144 (1) ◽  
pp. 71-86 ◽  
Author(s):  
Pierre Germain ◽  
Slim Ibrahim ◽  
Nader Masmoudi
Keyword(s):  
Maxwell Equations ◽  
Stokes Equations ◽  
Mild Solutions ◽  
A Priori ◽  
Local Existence ◽  
Large Data ◽  
Navier Stokes ◽  
Well Posedness ◽  
Scale Invariant ◽  
Navier Stokes Equations

We study the local and global well-posedness of a full system of magnetohydrodynamic equations. The system is a coupling of the incompressible Navier—Stokes equations with the Maxwell equations through the Lorentz force and Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale-invariant spaces classically used for Navier—Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions 2 and 3, but also refine those in [8]. The main simplification comes from an a prioriLt2 (Lx∞) estimate for solutions of the forced Navier—Stokes equations.

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