scholarly journals Semiflow selection for the compressible Navier–Stokes system

2020 ◽  
Author(s):  
Danica Basarić
Keyword(s):  
Initial Data ◽  
Open Problem ◽  
Stokes System ◽  
Initial Density ◽  
Finite Energy ◽  
Navier Stokes ◽  
State Variables ◽  
Semigroup Property ◽  
Almost Everywhere ◽  
Selection For

Abstract Although the existence of dissipative weak solutions for the compressible Navier–Stokes system has already been established for any finite energy initial data, uniqueness is still an open problem. The idea is then to select a solution satisfying the semigroup property, an important feature of systems with uniqueness. More precisely, we are going to prove the existence of a semiflow selection in terms of the three state variables: the density, the momentum, and the energy. Finally, we will show that it is possible to introduce a new selection defined only in terms of the initial density and momentum; however, the price to pay is that the semigroup property will hold almost everywhere in time.

Strong solutions to the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system

2019 ◽  
Vol 16 (04) ◽  
pp. 701-742 ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Stokes System ◽  
Initial Density ◽  
Local Existence ◽  
Strong Solutions ◽  
Incompressible Fluids ◽  
Navier Stokes ◽  
Initial Value ◽  
Two Phase ◽  
Density Dependent ◽  
Local Existence And Uniqueness

We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local existence and uniqueness of strong solutions to the initial value problem in a bounded domain, when the initial density function enjoys a positive lower bound.

scholarly journals Singularities of certain finite energy solutions to the Navier-Stokes system

2020 ◽  
Vol 40 (1) ◽  
pp. 189-206 ◽  
Author(s):  
Grzegorz Karch ◽  
◽  
Maria E. Schonbek ◽  
Tomas P. Schonbek ◽  
◽  
...  
Keyword(s):  
Stokes System ◽  
Finite Energy ◽  
Navier Stokes

The compressible Navier–Stokes–Cahn–Hilliard equations with dynamic boundary conditions

2019 ◽  
Vol 29 (14) ◽  
pp. 2557-2584 ◽  
Author(s):  
Laurence Cherfils ◽  
Eduard Feireisl ◽  
Martin Michálek ◽  
Alain Miranville ◽  
Madalina Petcu ◽  
...  
Keyword(s):  
Initial Data ◽  
Binary Mixture ◽  
Boundary Conditions ◽  
Finite Energy ◽  
Immiscible Fluids ◽  
Navier Stokes ◽  
Physical Domain ◽  
Dynamic Boundary Conditions ◽  
Dynamic Boundary ◽  
Rigid Walls

We consider the compressible Navier–Stokes–Cahn–Hilliard system describing the behavior of a binary mixture of compressible, viscous and macroscopically immiscible fluids. The equations are endowed with dynamic boundary conditions which allows taking into account the interaction between the fluid components and the rigid walls of the physical domain. We establish the existence of global-in-time weak solutions for any finite energy initial data.

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.

scholarly journals Markov selection for the stochastic compressible Navier–Stokes system

2020 ◽  
Vol 30 (6) ◽  
pp. 2547-2572
Author(s):  
Dominic Breit ◽  
Eduard Feireisl ◽  
Martina Hofmanová
Keyword(s):  
Stokes System ◽  
Navier Stokes ◽  
Markov Selection ◽  
Selection For

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 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}} .

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