scholarly journals On the Exponential Decay for Compressible Navier–Stokes–Korteweg Equations with a Drag Term

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
D. Bresch ◽  
M. Gisclon ◽  
I. Lacroix-Violet ◽  
A. Vasseur
Keyword(s):  
Exponential Decay ◽  
Navier Stokes ◽  
Drag Term

scholarly journals Feedback stabilization  of a 3D fluid flow by shape deformations of an obstacle

2021 ◽  
Author(s):  
Jean-Pierre Raymond ◽  
Muthusamy Vanninathan
Keyword(s):  
Fluid Flow ◽  
Decay Rate ◽  
Exponential Decay ◽  
Stokes Equations ◽  
Feedback Stabilization ◽  
Navier Stokes ◽  
Initial Shape ◽  
Exponential Decay Rate ◽  
Time Dependent Domain ◽  
Initial Location

We consider a fluid flow in a time dependent domain $\Omega_f(t)=\Omega \setminus \Omega_s(t)\subset {\mathbb R}^3$, surrounding a deformable obstacle $\Omega_s(t)$. We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in  $\Omega_f(t)$, $t>0$. We prove that, for any arbitrary exponential decay rate $\omega>0$, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary $\partial \Omega_s(t)$ can be chosen so that  the fluid flow is  stabilized to rest, and the obstacle to its initial shape and its initial location, with the  exponential decay rate $\omega>0$.

scholarly journals From Boltzmann to incompressible Navier–Stokes in Sobolev spaces with polynomial weight

2018 ◽  
Vol 17 (01) ◽  
pp. 85-116 ◽  
Author(s):  
Marc Briant ◽  
Sara Merino-Aceituno ◽  
Clément Mouhot
Keyword(s):  
Boltzmann Equation ◽  
Knudsen Number ◽  
Exponential Decay ◽  
Sobolev Spaces ◽  
Stokes Equations ◽  
Linear Part ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Global Equilibrium

We study the Boltzmann equation on the [Formula: see text]-dimensional torus in a perturbative setting around a global equilibrium under the Navier–Stokes linearization. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a [Formula: see text]-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently of the Knudsen number. Finally, we prove well-posedness of the Cauchy problem for the nonlinear Boltzmann equation in perturbative setting and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal. Furthermore, this result only requires derivatives in the space variable and allows to connect solutions to the incompressible Navier–Stokes equations in these spaces.

scholarly journals ON APPROXIMATE SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS II: GENERALIZATIONS, AND APPLICATIONS TO NAVIER–STOKES EQUATIONS

2008 ◽  
Vol 20 (06) ◽  
pp. 625-706 ◽  
Author(s):  
CARLO MOROSI ◽  
LIVIO PIZZOCCHERO
Keyword(s):  
Exact Solution ◽  
Global Existence ◽  
Exponential Decay ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Approximate Solutions ◽  
Navier Stokes ◽  
Small Data ◽  
Navier Stokes Equations ◽  
Semilinear Evolution Equations

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus Td of any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍn(Td), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a difference with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their ℍn distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

Density-induced CO2 dissolution - approaches to test a new hypothesis on a process relevant for epigenetic karstification

2020 ◽  
Author(s):  
Holger Class ◽  
Kilian Weishaupt ◽  
Oliver Trötschler ◽  
Harald Scherzer
Keyword(s):  
Wall Friction ◽  
Momentum Balance ◽  
Navier Stokes ◽  
Geologic Sequestration ◽  
Back Plate ◽  
Laboratory Flume ◽  
New Hypothesis ◽  
As Solubility ◽  
Color Indicator ◽  
Drag Term

<p><strong>Density-induced CO<sub>2</sub> dissolution - approaches to test a new hypothesis on a process relevant for epigenetic karstification   </strong></p><p>A process which has not yet been discussed as relevant for epigenetic karstification in phreatic zones has been hypothesized in a publication by Scherzer et al. (2017). It refers to an enhanced CO<sub>2</sub> transport into the phreatic zone by density-induced convective dissolution. The phenomenon is well-known also in CO<sub>2</sub> geologic sequestration and is denoted there typically as solubility trapping. Scherzer et al. (2017) denote this process in caves as nerochytic speleogenesis (from nerochytic = sink in Greek), assuming it has relevance for epigenetic karstification under certain circumstances. This could be relevant in particular in caves where CO<sub>2</sub> concentrations are highly elevated and show strong seasonal fluctuations.</p><p>Thomas et al. (2015) have introduced a method to visualize fingering patterns of CO<sub>2</sub> convective dissolution in water with a pH-sensitive color indicator. We have used this approach to produce a set of experimental data in a laboratory flume of dimensions 60 cm x 40 cm x 1 cm. Our aim is to validate a numerical model that we implemented in the simulator DuMu<sup>x </sup>(www.dumux.org), which can later on be used for future studies as the basis for investigating the relevance of nerochytic speleogenesis for karstification.</p><p>We have applied atmospheres with varying concentrations of carbon dioxid as boundary conditions at the top of the flume and observed the onset times and fingering patterns, in particular we focused on the velocity of the fingers.</p><p>The Navier-Stokes model with water density dependent on CO<sub>2</sub> concentration is run in 2D, 3D and pseudo 3D, the latter referring to a 2D approach with a drag term in the momentum balance to account for wall friction at the front and the back plate. Without calibration or fitting of parameters, the results of the comparison between experiment and simulation show reasonable agreement both with respect to the onset of convective fingering and the number of fingers occurring.</p><p>References:</p><p>H. Scherzer, H. Class, K. Weishaupt, T. Sauerborn, O. Trötschler: Nerochytische Speläogenese: Konvektiver Vertikaltransport von gelöstem CO<sub>2</sub> - ein Antrieb für Verkarstung in der phreatischen Zone im Bedeckten Karst, Laichinger Höhlenfreund 52:29-35, ISSN 0344 6832, 2017.<br> </p><p>C. Thomas, L. Lemaigre, A. Zalts, A. D'Onofrio, A. De Wit: Experimental study of CO<sub>2</sub> convective dissolution: the effect of color indicators, International Journal of Greenhouse Gas Control 42:525-533,2015.</p>

scholarly journals EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS II: HOOKEAN-TYPE MODELS

2012 ◽  
Vol 22 (05) ◽  
pp. 1150024 ◽  
Author(s):  
JOHN W. BARRETT ◽  
ENDRE SÜLI
Keyword(s):  
Center Of Mass ◽  
Planck Equation ◽  
Momentum Equation ◽  
Polymer Chains ◽  
Navier Stokes ◽  
Extra Stress Tensor ◽  
The Right ◽  
Extra Stress ◽  
Drag Term ◽  
Fokker Planck

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0 for the Navier–Stokes equation and a non-negative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (with respect to the measure [Formula: see text]) over any time interval [0, T], T>0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text]-norm, at a rate that is independent of (ṵ0, ψ0) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the Kolmogorov–Riesz theorem to Banach-space-valued Sobolev spaces.

scholarly journals Global dissipative solutions of the defocusing isothermal Euler–Langevin–Korteweg equations

2021 ◽  
pp. 1-29
Author(s):  
Quentin Chauleur
Keyword(s):  
Weak Solutions ◽  
Type Inequality ◽  
Quantum Fluids ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Global Weak Solutions ◽  
Dissipative Solutions ◽  
Linear Drag ◽  
Standard Approximation ◽  
Drag Term

We construct global dissipative solutions on the torus of dimension at most three of the defocusing isothermal Euler–Langevin–Korteweg system, which corresponds to the Euler–Korteweg system of compressible quantum fluids with an isothermal pressure law and a linear drag term with respect to the velocity. In particular, the isothermal feature prevents the energy and the BD-entropy from being positive. Adapting standard approximation arguments we first show the existence of global weak solutions to the defocusing isothermal Navier–Stokes–Langevin–Korteweg system. Introducing a relative entropy function satisfying a Gronwall-type inequality we then perform the inviscid limit to obtain the existence of dissipative solutions of the Euler–Langevin–Korteweg system.

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