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2021 | Book

Semigroup Theory for the Stokes Operator with Navier Boundary Condition on Lp Spaces Read chapter Read first chapter

Waves in Flows

The 2018 Prague-Sum Workshop Lectures

Editors: Dr. Tomáš Bodnár, Prof. Giovanni P. Galdi, Šárka Nečasová

Publisher: Springer International Publishing

Book Series : Advances in Mathematical Fluid Mechanics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume explores a range of recent advances in mathematical fluid mechanics, covering theoretical topics and numerical methods. Chapters are based on the lectures given at a workshop in the summer school Waves in Flows, held in Prague from August 27-31, 2018. A broad overview of cutting edge research is presented, with a focus on mathematical modeling and numerical simulations. Readers will find a thorough analysis of numerous state-of-the-art developments presented by leading experts in their respective fields. Specific topics covered include:

ChemorepulsionCompressible Navier-Stokes systemsNewtonian fluidsFluid-structure interactions

Waves in Flows: The 2018 Prague-Sum Workshop Lectures will appeal to post-doctoral students and scientists whose work involves fluid mechanics.

Table of Contents

Frontmatter
Chapter 1. Semigroup Theory for the Stokes Operator with Navier Boundary Condition on Lp Spaces
Abstract
We consider the incompressible Navier–Stokes equations in a bounded domain with \(\mathcal {C}^{1,1}\) boundary, completed with slip boundary condition. We study the general semigroup theory in L p-spaces related to the Stokes operator with Navier boundary condition where the slip coefficient α is a non-smooth scalar function. It is shown that the strong and weak Stokes operators with Navier conditions admit analytic semigroup with bounded pure imaginary powers. We also show that for α large, the weak and strong solutions of both the linear and nonlinear systems are bounded uniformly with respect to α. This justifies mathematically that the solution of the Navier–Stokes problem with slip condition converges in the energy space to the solution of the Navier–Stokes with no-slip boundary condition as α →∞.
Chérif Amrouche, Miguel Escobedo, Amrita Ghosh
Chapter 2. Theoretical and Numerical Results for a Chemorepulsion Model with Non-constant Diffusion Coefficients
Abstract
Chemotaxis is the biological process of the movement of living organisms in response to a chemical stimulus that can be given toward a higher (attractive) or lower (repulsive) concentration of a chemical substance. At the same time, the presence of living organisms can produce or consume chemical substance.
Francisco Guillén-González, María Ángeles Rodríguez-Bellido, Diego Armando Rueda-Gómez
Chapter 3. Remarks on the Energy Equality for the 3D Navier-Stokes Equations
Abstract
In a recent paper, jointly with L. C. Berselli, we study the problem of energy conservation for solutions of the initial boundary value problem associated with the 3D Navier-Stokes equations with Dirichlet boundary conditions. While the energy equality is satisfied for strong solutions, the dissipation phenomenon is expected to be connected with the roughness of the solutions. A natural question is, then, which regularity is needed for a weak solution in order to conserve the energy. The importance of this issue was brought out in evidence by Onsager’s work.
Luigi Carlo Berselli, Elisabetta Chiodaroli
Chapter 4. Existence, Uniqueness, and Asymptotic Behavior of Regular Time-Periodic Viscous Flow Around a Moving Body
Abstract
We show existence and uniqueness of regular time-periodic solutions to the Navier–Stokes problem in the exterior of a rigid body, \(\mathcal B\), that moves by arbitrary (sufficiently smooth) time-periodic translational motion of the same period, provided the size of the data is suitably restricted. Moreover, we characterize the spatial asymptotic behavior of such solutions and prove, in particular, that if \(\mathcal B\) has a nonzero net motion identified by a constant velocity \(\overline {\boldsymbol {\xi }}\) (say), then the solution exhibits a wake-like behavior in the direction \(-\overline {\boldsymbol {\xi }}\) entirely analogous to that of a steady-state flow around a body that moves with velocity \(\overline {\boldsymbol {\xi }}\).
Giovanni Paolo Galdi
Chapter 5. Compressible Navier-Stokes System on a Moving Domain in the Lp − Lq Framework
Abstract
We prove the local well-posedness for the barotropic compressible Navier-Stokes system on a moving domain, a motion of which is determined by a given vector field V, in a maximal L p − L q regularity framework. Under additional smallness assumptions on the data we show that our solution exists globally in time and satisfies a decay estimate. In particular, for the global well-posedness we do not require exponential decay or smallness of V in L p(L q). However, we require exponential decay and smallness of its derivatives.
Ondřej Kreml, Šárka Nečasová, Tomasz Piasecki
Chapter 6. Some New Properties of a Suitable Weak Solution to the Navier–Stokes Equations
Abstract
The paper is concerned with the IBVP of the Navier–Stokes equations. The goal is the construction of a weak solution enjoying some new properties. Of course, we look for properties that are global in time. The results hold assuming an initial data v 0 ∈ J 2(Ω).
Francesca Crispo, Paolo Maremonti, Carlo Romano Grisanti
Chapter 7. Existence, Uniqueness, and Regularity for the Second–Gradient Navier–Stokes Equations in Exterior Domains
Abstract
We study the well-posedness of the problem
$$\displaystyle \left \{\begin {array}{ll} \dfrac {\partial u}{\partial t}+\left (Du\right )u+\nabla p =\nu \varDelta u-\tau \varDelta \varDelta u & \mathrm {in} \ ]0,+\infty [\times \varOmega \,,\\ \operatorname {div} u=0 & \mathrm {in} \ ]0,+\infty [\times \varOmega \,,\\ u(t,x)=\dfrac {\partial u}{\partial n}(t,x)=0 & \mathrm {on} \ ]0,+\infty [\times \partial \varOmega \,,\\ u(0,x)=u_0(x) & \mathrm {in} \ \varOmega \,, \end {array} \right . $$
where https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-68144-9_7/485873_1_En_7_IEq1_HTML.gif is the velocity field, https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-68144-9_7/485873_1_En_7_IEq2_HTML.gif is the pressure, ν is the kinematical viscosity, τ the so-called hyperviscosity and Ω is a general domain as for existence and uniqueness of the solution, and an exterior domain as for regularity results.
Marco Degiovanni, Alfredo Marzocchi, Sara Mastaglio
Chapter 8. A Review on Rigorous Derivation of Reduced Models for Fluid–Structure Interaction Systems
Abstract
In this paper we review and systematize the mathematical theory on justification of sixth-order thin-film equations as reduced models for various fluid–structure interaction systems in which fluids are lubricating underneath elastic structures. Justification is based on careful examination of energy estimates, weak convergence results of solutions of the original fluid–structure interaction systems to the solution of the sixth-order thin-film equation, and quantitative error estimates that provide even strong convergence results.
Mario Bukal, Boris Muha
Chapter 9. Stability of a Steady Flow of an Incompressible Newtonian Fluid in an Exterior Domain
Abstract
This chapter provides a brief survey of studies on stability of a steady flow of an incompressible Newtonian fluid around a compact body \(\mathcal {B}\). Results on the long-time behavior and stability under assumptions of “sufficient smallness” of some quantities are cited and briefly described in Sect. 9.2. Results, mainly based on assumptions on spectrum of a certain associated linear operator are presented in Sect. 9.3. Finally, Sect. 9.4 contains a short note on analogous results concerning the case when body \(\mathcal {B}\) rotates.
Jiří Neustupa
Metadata
Title
Waves in Flows
Editors
Dr. Tomáš Bodnár
Prof. Giovanni P. Galdi
Šárka Nečasová
Copyright Year
2021
Electronic ISBN
978-3-030-68144-9
Print ISBN
978-3-030-68143-2
DOI
https://doi.org/10.1007/978-3-030-68144-9

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