Waves in Flows The 2018 Prague-Sum Workshop Lectures

Preface
Contents
1 Semigroup Theory for the Stokes Operator with NavierBoundary Condition on Lp Spaces
	1.1 Introduction
	1.2 Preliminaries
	1.3 Stokes Operator on Lpσ,τ(Ω): Strong Solutions
	1.4 Stokes Operator on [H0p(div,Ω)]σ, τ
	1.5 Imaginary and Fractional Powers
		1.5.1 Imaginary Powers
		1.5.2 Fractional Powers
	1.6 The Homogeneous Stokes Problem
		1.6.1 Strong Solution
		1.6.2 Weak Solution
	1.7 The Non-homogeneous Stokes Problem
	1.8 Nonlinear Problem
	1.9 The Limit as α→∞
	References
2 Theoretical and Numerical Results for a ChemorepulsionModel with Non-constant Diffusion Coefficients
	2.1 Introduction
	2.2 Chemorepulsion Production System with Quadratic Production Term
	2.3 Existence of Weak Solution
		2.3.1 First Weak Estimates
		2.3.2 Some Additional Properties
		2.3.3 Additional Weak Estimate for v
	2.4 Uniqueness Criteria for Weak Solutions
	2.5 Some Constraints Implying Strong Estimates
	2.6 More Regular Estimates
	2.7 An Equivalent Problem (σ=v)
		2.7.1 The Stationary Problem for σ
		2.7.2 Weak Regularity of the (u, σ) Problem (2.32)
		2.7.3 Equivalence of Problems (u,v) and (u,σ)
	2.8 Asymptotic Analysis
	2.9 Time-Discrete Schemes
		2.9.1 Main Properties of the Time-Discrete Schemes
		2.9.2 A Linear Scheme
	2.10 Fully Discrete Schemes
		2.10.1 Large-Time Behavior
	2.11 Numerical Simulations
	2.12 Conclusions
	References
3 Remarks on the Energy Equality for the 3D Navier-StokesEquations
	3.1 Introduction
	3.2 Main Results
	3.3 Comparison with Previous Results
		3.3.1 Energy Conservation and Onsager Conjecture
		3.3.2 The Case q<3
		3.3.3 On the Shinbrot Condition (3.3) and Very-WeakSolutions
	References
4 Existence, Uniqueness, and Asymptotic Behavior of RegularTime-Periodic Viscous Flow Around a Moving Body
	4.1 Introduction
	4.2 Preliminaries
	4.3 On the Unique Solvability of the Linear Problem
	4.4 On the Unique Solvability of the NonlinearProblem
	References
5 Compressible Navier-Stokes System on a Moving Domainin the Lp-Lq Framework
	5.1 Introduction
		5.1.1 Notation
		5.1.2 Main Results
	5.2 Lagrangian Transformation
	5.3 Linear Theory and Auxiliary Results
		5.3.1 Lp-Lq Maximal Regularity
		5.3.2 Exponential Decay
			Linearization
			Proof of Propositions 5.3.2 and 5.3.3
		5.3.3 Embedding Results
	5.4 Local Well-Posedness
		5.4.1 Linearization for the Local Well-Posedness
		5.4.2 Nonlinear Estimates for the Local Well-Posedness
		5.4.3 Fixed Point Argument
		5.4.4 Equivalence of Norms in Lagrangian and EulerianCoordinates
	5.5 Global Well-Posedness
		5.5.1 Linearization
		5.5.2 Nonlinear Estimates for the Global Well-Posedness
		5.5.3 Proof of Theorem 5.1.2
	References
6 Some New Properties of a Suitable Weak Solution to theNavier–Stokes Equations
	6.1 Introduction
	6.2 Some Preliminary Results
	6.3 Proof of Theorem 6.1.1
		6.3.1 The Strong Convergence in Lp(0,T;L2(Ω))for All p[1,2)
		6.3.2 Proof of Formula (6.2)
		6.3.3 Proof of Formula (6.3)
		6.3.4 The Lμ(q)(0,T;Lq(Ω)) Property
	6.4 Appendix
		6.4.1 Some Results Related to the Construction of the WeakSolution
		6.4.2 Uniqueness Backward in Time for the Sequence {vm}
	References
7 Existence, Uniqueness, and Regularity for theSecond–Gradient Navier–Stokes Equations in ExteriorDomains
	7.1 Introduction
	7.2 Notations and Preliminaries
		7.2.1 Sobolev Spaces
		7.2.2 Hydrodynamic Spaces
	7.3 The Nonlinear Term
	7.4 Steady-State Variational Inequalities
	7.5 Uniqueness and Existence
	7.6 Regularity in Space for Exterior Domains
	References
8 A Review on Rigorous Derivation of Reduced Modelsfor Fluid–Structure Interaction Systems
	8.1 Introduction
	8.2 Heuristic Derivation of Reduced Models
	8.3 Reduced Models for Fluid–Structure Interaction Problems:A Rigorous Approach
		8.3.1 Linear + 0 Problem
		8.3.2 Linear + h Problem
		8.3.3 Nonlinear + 0 Problem
	References
9 Stability of a Steady Flow of an IncompressibleNewtonian Fluid in an Exterior Domain
	9.1 Introduction
	9.2 Long-Time Behavior and Stability Under Assumptions of Smallness of Some Data
	9.3 Spectral Methods
	9.4 A Note on Stability of a Steady Flow Past a Rotating Body
	References