scholarly journals Existence and Uniqueness of Isothermal, Slightly Compressible Stratified Flow

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Arianna Passerini
Keyword(s):  
Existence And Uniqueness ◽  
Compressible Fluids ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Small Data ◽  
Well Posedness ◽  
Balance Laws ◽  
Regular Solutions ◽  
Arbitrary Size ◽  
Slightly Compressible

AbstractWe show well-posedness for the equations describing a new model of slightly compressible fluids. This model was recently rigorously derived in Grandi and Passerini (Geophys Astrophys Fluid Dyn, 2020) from the full set of balance laws and falls in the category of anelastic Navier–Stokes fluids. In particular, we prove existence and uniqueness of global regular solutions in the two-dimensional case for initial data of arbitrary “size”, and for “small” data in three dimensions. We also show global stability of the rest state in the class of weak solutions.

scholarly journals Bénard Problem for Slightly Compressible Fluids: Existence and Nonlinear Stability in 3D

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Arianna Passerini
Keyword(s):  
Pressure Field ◽  
Fluid Model ◽  
Classical Model ◽  
Compressible Fluids ◽  
Heat Conducting ◽  
Regular Solutions ◽  
New Approach ◽  
Slightly Compressible ◽  
Bénard Problem ◽  
Benard Problem

This paper shows the existence, uniqueness, and asymptotic behavior in time of regular solutions (a la Ladyzhenskaya) to the Bénard problem for a heat-conducting fluid model generalizing the classical Oberbeck–Boussinesq one. The novelty of this model, introduced by Corli and Passerini, 2019, and Passerini and Ruggeri, 2014, consists in allowing the density of the fluid to also depend on the pressure field, which, as shown by Passerini and Ruggeri, 2014, is a necessary request from a thermodynamic viewpoint when dealing with convective problems. This property adds to the problem a rather interesting mathematical challenge that is not encountered in the classical model, thus requiring a new approach for its resolution.

scholarly journals On well-posedness of a magnetization-variables model for Navier-Stokes equations with damping

2020 ◽  
Author(s):  
Abdelkerim Chaabani,
Keyword(s):  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Well Posedness ◽  
Energy Methods ◽  
Damping Term ◽  
Navier Stokes Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results

This paper aims to establish existence and uniqueness results of weak and strong solution to the three-dimensional periodic magnetization-variables formulation to Navier-Stokes equations with damping term. Authors in precedent works addressed the question as to whether this model and similar ones without damping term possess a weak solution. In this vein, considering a damping term in the magnetization-variable formulation turned to be consequential as it enforces existence and uniqueness results. Energy methods, compactness methods are the main tools.

scholarly journals Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis

2017 ◽  
Vol 29 (4) ◽  
pp. 595-644 ◽  
Author(s):  
KEI FONG LAM ◽  
HAO WU
Keyword(s):  
Mass Transfer ◽  
Chemical Potential ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Fluid Viscosity ◽  
Well Posedness ◽  
Global Weak Solutions ◽  
Two Phase ◽  
Model Variant

We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

scholarly journals Well-posedness of evolutionary Navier-Stokes equations with forces of low regularity on two-dimensional domains

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Eduardo Renteria Casas
Keyword(s):  
Stokes Equation ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Well Posedness ◽  
Uniqueness Of Solutions ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Arbitrary Dimensions ◽  
Parameter Ranges

Existence and uniqueness of solutions to the Navier-Stokes equation in dimension two with forces in the space $L^q( (0,T); \bWmop)$ for $p$ and $q$ in  appropriate parameter ranges are proven. The case of spatially measured-valued inhomogeneities is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions with $1 < p, q < \infty$ arbitrary.

Well-posedness of the time-space fractional stochastic Navier-Stokes equations driven by fractional Brownian motion

2018 ◽  
Vol 13 (1) ◽  
pp. 11 ◽  
Author(s):  
Pengfei Xu ◽  
Caibin Zeng ◽  
Jianhua Huang
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Banach Fixed Point Theorem ◽  
Well Posedness ◽  
Stochastic Convolution ◽  
Navier Stokes Equations ◽  
Time Space

The current paper is devoted to the time-space fractional Navier-Stokes equations driven by fractional Brownian motion. The spatial-temporal regularity of the nonlocal stochastic convolution is firstly established, and then the existence and uniqueness of mild solution are obtained by Banach Fixed Point theorem and Mittag-Leffler families operators.

Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems

2021 ◽  
pp. 1-32
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Three Dimensional ◽  
Decay Rates ◽  
Navier Stokes ◽  
Small Data ◽  
Well Posedness ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
The Cauchy Problem ◽  
Time Decay Rates ◽  
Derivatives Of

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$ ( $0\leq s<\frac {3}{2}$ ) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

scholarly journals Notes on the Global Well-Posedness for the Maxwell-Navier-Stokes System

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ensil Kang ◽  
Jihoon Lee
Keyword(s):  
Global Existence ◽  
Stokes System ◽  
Blow Up ◽  
Energy Estimates ◽  
Navier Stokes ◽  
Well Posedness ◽  
Regular Solutions ◽  
2D System ◽  
Blow Up Criterion

Masmoudi (2010) obtained global well-posedness for 2D Maxwell-Navier-Stokes system. In this paper, we reprove global existence of regular solutions to the 2D system by using energy estimates and Brezis-Gallouet inequality. Also we obtain a blow-up criterion for solutions to 3D Maxwell-Navier-Stokes system.

scholarly journals On the Solvability of a Problem of Stationary Fluid Flow in a Helical Pipe

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Igor Pažanin
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Fluid Flow ◽  
Existence And Uniqueness ◽  
Stokes System ◽  
Navier Stokes ◽  
Small Data ◽  
Helical Pipe ◽  
Pressure Boundary Condition ◽  
Incompressible Newtonian Fluid

We consider a flow of incompressible Newtonian fluid through a pipe with helical shape. We suppose that the flow is governed by the prescribed pressure drop between pipe's ends. Such model has relevance to some important engineering applications. Under small data assumption, we prove the existence and uniqueness of the weak solution to the corresponding Navier-Stokes system with pressure boundary condition. The proof is based on the contraction method.

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