Global existence of weak solution to compressible Navier-Stokes/Allen-Cahn system in three dimensions

2019 ◽  
Vol 477 (2) ◽  
pp. 1265-1295 ◽  
Author(s):  
Senming Chen ◽  
Huanyao Wen ◽  
Changjiang Zhu
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Three Dimensions ◽  
Navier Stokes

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.

scholarly journals Global existence for a two-species chemotaxis-Navier-Stokes system with $ p $-Laplacian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jiayi Han ◽  
Changchun Liu
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Stokes System ◽  
Three Dimensional ◽  
Global Weak Solution ◽  
Navier Stokes ◽  
Bounded Domains

<p style='text-indent:20px;'>We consider a two-species chemotaxis-Navier-Stokes system with <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian in three-dimensional smooth bounded domains. It is proved that for any <inline-formula><tex-math id="M3">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>, the problem admits a global weak solution.</p>

scholarly journals On a Bingham fluid whose viscosity and yield limit depend on the temperature

1992 ◽  
Vol 128 ◽  
pp. 1-14 ◽  
Author(s):  
Yoshio Kato
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Initial Velocity ◽  
Local Existence ◽  
Strong Solutions ◽  
Bingham Fluid ◽  
Affirmative Answer ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Yield Limit

Duvaut and Lions [2] studied the field of velocities and of temperatures in a moving incompressible Bingham fluid endowed with viscosity μ(θ) depending on the temperature θ and established the existence of a weak solution in the case of a two dimensional fluid. However, the problem of uniqueness remained unsolved. The purpose of the present paper is to give an affirmative answer to the problem, that is, to show the local existence (resp. the global existence) in the time and the uniqueness of (strong) solutions in three dimensions under the conditions that (i) the time (resp. the initial velocity and the external force) and (ii) the rate of variation of the viscosity and the yield limit with respect to the temperature are both sufficiently small. It will be easily seen that the global existence and the uniqueness also hold in the two dimensional case whenever the rate (ii) is sufficiently small.

scholarly journals On the stationary nonlocal Cahn–Hilliard–Navier–Stokes system: Existence, uniqueness and exponential stability

2020 ◽  
pp. 1-41
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan ◽  
Lakshmi Naga Mahendranath Perisetti
Keyword(s):  
Weak Solution ◽  
Stokes System ◽  
Two Dimensions ◽  
Immiscible Fluids ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Browder’S Theorem ◽  
Exponentially Stable ◽  
Mobility Parameter ◽  
Regularity Results

The Cahn–Hilliard–Navier–Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn–Hilliard–Navier–Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder’s theorem. Further, we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn–Hilliard–Navier–Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable (for convex singular potentials) under suitable conditions on mobility parameter and viscosity.

Singularities

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Stokes Equations ◽  
Negative Answer ◽  
Three Dimensions ◽  
Picard Iteration ◽  
Navier Stokes ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
L2 Norm ◽  
Two Dimensional Flow ◽  
Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

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