INHOMOGENEOUS INCOMPRESSIBLE VISCOUS FLOWS WITH SLOWLY VARYING INITIAL DATA

2016 ◽  
Vol 17 (5) ◽  
pp. 1121-1172
Author(s):  
Jean-Yves Chemin ◽  
Ping Zhang
Keyword(s):  
Initial Data ◽  
Stokes System ◽  
Large Data ◽  
Viscous Flows ◽  
Approximate Solutions ◽  
Navier Stokes ◽  
Decay Properties ◽  
Optimal Decay ◽  
Whole Space ◽  
Well Posed

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3D inhomogeneous incompressible Navier–Stokes system in the whole space $\mathbb{R}^{3}$. This class of data is based on functions which vary slowly in one direction. The idea is that 2D inhomogeneous Navier–Stokes system with large data is globally well-posed and we construct the 3D approximate solutions by the 2D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2D inhomogeneous Navier–Stokes system. We obtained the same optimal decay estimates as the solutions of 2D homogeneous Navier–Stokes system.

scholarly journals Linear stability of blowup solution of incompressible Keller–Segel–Navier–Stokes system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan Yan ◽  
Hengyan Li
Keyword(s):  
Initial Data ◽  
Linear Stability ◽  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Blowup Solution ◽  
Existence Time ◽  
Whole Space

AbstractIn this paper, we consider the linear stability of blowup solution for incompressible Keller–Segel–Navier–Stokes system in whole space $\mathbb{R}^{3}$ R 3 . More precisely, we show that, if the initial data of the three dimensional Keller–Segel–Navier–Stokes system is close to the smooth initial function $(0,0,\textbf{u}_{s}(0,x) )^{T}$ ( 0 , 0 , u s ( 0 , x ) ) T , then there exists a blowup solution of the three dimensional linear Keller–Segel–Navier–Stokes system satisfying the decomposition $$ \bigl(n(t,x),c(t,x),\textbf{u}(t,x) \bigr)^{T}= \bigl(0,0, \textbf{u}_{s}(t,x) \bigr)^{T}+\mathcal{O}(\varepsilon ), \quad \forall (t,x)\in \bigl(0,T^{*}\bigr) \times \mathbb{R}^{3}, $$ ( n ( t , x ) , c ( t , x ) , u ( t , x ) ) T = ( 0 , 0 , u s ( t , x ) ) T + O ( ε ) , ∀ ( t , x ) ∈ ( 0 , T ∗ ) × R 3 , in Sobolev space $H^{s}(\mathbb{R}^{3})$ H s ( R 3 ) with $s=\frac{3}{2}-5a$ s = 3 2 − 5 a and constant $0< a\ll 1$ 0 < a ≪ 1 , where $T^{*}$ T ∗ is the maximal existence time, and $\textbf{u}_{s}(t,x)$ u s ( t , x ) given in (Yan 2018) is the explicit blowup solution admitted smooth initial data for three dimensional incompressible Navier–Stokes equations.

FLOWS OF VISCOUS COMPRESSIBLE FLUIDS UNDER STRONG STRATIFICATION: INCOMPRESSIBLE LIMITS FOR LONG-RANGE POTENTIAL FORCES

2011 ◽  
Vol 21 (01) ◽  
pp. 7-27 ◽  
Author(s):  
EDUARD FEIREISL
Keyword(s):  
Mach Number ◽  
Acoustic Waves ◽  
Stokes System ◽  
Hybrid Methods ◽  
Singular Limit ◽  
Acoustic Energy ◽  
Compressible Fluids ◽  
Navier Stokes ◽  
Whole Space ◽  
Rage Theorem

We study the singular limit of the compressible Navier–Stokes system in the whole space ℝ3, where the Mach number and Froude number are proportional to a small parameter ε → 0. The central issue is the local decay of the acoustic energy proved by means of the RAGE theorem. The result is quite general and the proposed approach can be applied to a large variety of problems that concern propagation of acoustic waves in compressible fluids. In particular, the method can be used for showing stability of various numerical schemes based on the so-called hybrid methods.

scholarly journals ON COUPLED PROBLEMS FOR VISCOUS FLOW IN EXTERIOR DOMAINS

1998 ◽  
Vol 08 (04) ◽  
pp. 657-684 ◽  
Author(s):  
M. FEISTAUER ◽  
C. SCHWAB
Keyword(s):  
Stokes Flow ◽  
Stokes System ◽  
Stokes Problem ◽  
Large Data ◽  
Transmission Conditions ◽  
Navier Stokes ◽  
Coupled Problems ◽  
Exterior Domains ◽  
Existence Of A Solution ◽  
Coupled Problem

The use of the complete Navier–Stokes system in an unbounded domain is not always convenient in computations and, therefore, the Navier–Stokes problem is often truncated to a bounded domain. In this paper we simulate the interaction between the flow in this domain and the exterior flow with the aid of a coupled problem. We propose in particular a linear approximation of the exterior flow (here the Stokes flow or potential flow) coupled with the interior Navier–Stokes problem via suitable transmission conditions on the artificial interface between the interior and exterior domains. Our choice of the transmission conditions ensures the existence of a solution of the coupled problem, also for large data.

Singular limit of a dispersive Navier–Stokes system with an entropy structure

2014 ◽  
Vol 13 (01) ◽  
pp. 77-99
Author(s):  
C. David Levermore ◽  
Weiran Sun ◽  
Konstantina Trivisa
Keyword(s):  
Stokes System ◽  
Singular Limit ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Third Order ◽  
Fluid System ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Whole Space ◽  
Entropy Structure

We prove a low Mach number limit for a dispersive fluid system [3] which contains third-order corrections to the compressible Navier–Stokes. We show that the classical solutions to this system in the whole space ℝn converge to classical solutions to ghost-effect systems [7]. Our analysis follows the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and to simplify the analysis in [4].

scholarly journals GLOBAL SOLUTION FOR THE GENERALIZED ANISOTROPIC NAVIER–STOKES EQUATIONS WITH LARGE DATA

2015 ◽  
Vol 20 (2) ◽  
pp. 205-231 ◽  
Author(s):  
Daoyuan Fang ◽  
Bin Han
Keyword(s):  
Initial Data ◽  
Besov Space ◽  
Global Solution ◽  
Stokes Equations ◽  
Large Data ◽  
Navier Stokes ◽  
Unique Global Solution ◽  
Large Initial Data ◽  
Navier Stokes Equations ◽  
Anisotropic Besov Space

We are concerned with 3D incompressible generalized anisotropic Navier– Stokes equations with hyperdissipative term in horizontal variables. We prove that there exists a unique global solution for it with large initial data in anisotropic Besov space.

scholarly journals Self-propelled motion of a rigid body inside a density dependent incompressible fluid

2020 ◽  
Author(s):  
Sarka Necasova ◽  
Mythily Ramaswamy ◽  
Arnab Roy ◽  
Anja Schlomerkemper
Keyword(s):  
Angular Momentum ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Rigid Body ◽  
Incompressible Fluid ◽  
Stokes System ◽  
Navier Stokes ◽  
Navier Condition ◽  
Whole Space

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole space. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the  balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.

Existence of global weak solution for quantum Navier–Stokes system

2020 ◽  
Vol 31 (05) ◽  
pp. 2050038
Author(s):  
Jianwei Yang ◽  
Gaohui Peng ◽  
Huiyun Hao ◽  
Fengzhen Que
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Large Data ◽  
Global Weak Solution ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Dependent Viscosity ◽  
Time Existence ◽  
Cold Pressure

In this paper, the barotropic compressible quantum Navier–Stokes equations with a density-dependent viscosity in a three-dimensional torus is studied. By introducing a cold pressure to handle the convection term, we prove the global-in-time existence of weak solutions to quantum Navier–Stokes equations for large data in the sense of standard definition.

scholarly journals Suppressing blow-up by gradient-dependent flux limitation in a planar Keller–Segel–Navier–Stokes system

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Michael Winkler
Keyword(s):  
Dirichlet Problem ◽  
Initial Data ◽  
Classical Solution ◽  
Smooth Function ◽  
Stokes System ◽  
Blow Up ◽  
Navier Stokes ◽  
Diffusive Fluxes ◽  
Fluid Interaction ◽  
Flux Limitation

AbstractThe flux-limited Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ n t + u · ∇ n = Δ n - ∇ · ( n f ( | ∇ c | 2 ) ∇ c ) , c t + u · ∇ c = Δ c - c + n , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , ( ⋆ ) is considered in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . It is shown that whenever the suitably smooth function f models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that $$\begin{aligned} |f(\xi )| \le K_f\cdot (\xi +1)^{-\frac{\alpha }{2}} \end{aligned}$$ | f ( ξ ) | ≤ K f · ( ξ + 1 ) - α 2 holds for all $$\xi \ge 0$$ ξ ≥ 0 with some $$K_f>0$$ K f > 0 and $$\alpha >0$$ α > 0 , for any all reasonably regular initial data a corresponding no-flux/no-flux/Dirichlet problem admits a globally defined classical solution which is bounded, inter alia, in $$L^\infty (\Omega \times (0,\infty ))$$ L ∞ ( Ω × ( 0 , ∞ ) ) with respect to all its components. By extending a corresponding result known for a fluid-free counterpart of ($$\star $$ ⋆ ), this confirms that with regard to the possible emergence of blow-up phenomena, the choice $$f\equiv const.$$ f ≡ c o n s t . retains some criticality also in the presence of fluid interaction.

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