Recent Developments of Mathematical Fluid Mechanics

Table of Contents

1. Frontmatter

2. The Work of Yoshihiro Shibata

   Herbert Amann, Yoshikazu Giga, Hisashi Okamoto, Hideo Kozono, Masaso Yamazaki

   Abstract

   Introducing his research carrier, we address Prof. Yoshihiro Shibata’s great contributions to the mathematical analysis. His out-standing influence to the mathematical society is also clarified.

3. Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type Two-Phase Flows

   Helmut Abels, Lars Diening, Yutaka Terasawa

   Abstract

   We first review results about existence of generalized or weak solutions for Newtonian and power-law type two-phase flows. Then we state a recent result by the authors about existence of weak solutions for diffuse interface model of power-law type two-phase flows and give a sketch of its proof. The latter part is a summary of Abels et al. (Nonlinear Anal Real World Appl 15:149–157, 2014).

4. Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free Energies

   Helmut Abels, Josef Weber

   Abstract

   We consider a stationary Navier-Stokes/Cahn-Hilliard type system. The system describes a so-called diffuse interface model for the two-phase flow of two macroscopically immiscible incompressible viscous fluids in the case of matched densities, also known as Model H. We prove existence of weak solutions for the stationary system for general exterior forces and singular free energies, which ensure that the order parameter stays in the physical reasonable interval. To this end we reduce the system to an abstract differential inclusion and apply the theory of multi-valued pseudo-monotone operators.

5. Parabolic Equations on Uniformly Regular Riemannian Manifolds and Degenerate Initial Boundary Value Problems

   Herbert Amann

   Abstract

   In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary.

6. A Generalization of Some Regularity Criteria to the Navier–Stokes Equations Involving One Velocity Component

   Šimon Axmann, Milan Pokorný

   Abstract

   We present generalizations of results concerning conditional global regularity of weak Leray–Hopf solutions to incompressible Navier–Stokes equations presented by Zhou and Pokorný in articles (Pokorný, Electron J Differ Equ (11):1–8, 2003; Zhou, Methods Appl Anal 9(4):563–578, 2002; Zhou, J Math Pure Appl 84(11):1496–1514, 2005); see also Neustupa et al. (Quaderni di Matematica, vol. 10. Topics in Mathematical Fluid Mechanics, 2002, pp. 163–183) We are able to replace the condition on one velocity component (or its gradient) by a corresponding condition imposed on a projection of the velocity (or its gradient) onto a more general vector field. Comparing to our other recent results from Axmann and Pokorný (A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component, in preparation), the conditions imposed on the projection are more restrictive here, however due to the technique used in Axmann and Pokorný (A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component, in preparation), there appeared a specific additional restriction on geometrical properties of the reference field, which could be omitted here.

7. On the Singular p-Laplacian System Under Navier Slip Type Boundary Conditions: The Gradient-Symmetric Case

   H. Beirão da Veiga

   Abstract

   We consider the p-Laplacian system of N equations in n space variables, 1 < p ≤ 2 , under the homogeneous Navier slip boundary condition without friction. Here, the gradient of the velocity is replaced by the (more physical) symmetric gradient, and the classical non-slip boundary condition is replaced by the Navier slip boundary condition without friction. These combination of circumstances leads to some additional obstacles. We prove W ^2, q  regularity, up to the boundary, under suitable assumptions on the couple p, q . The singular case μ = 0 is covered.

8. Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles in an Incompressible Solvent

   Dieter Bothe, Kohei Soga

   Abstract

   We derive mathematical models of the elementary process of dissolution/growth of bubbles in a liquid under pressure control. The modeling starts with a fully compressible version, both for the liquid and the gas phase so that the entropy principle can be easily evaluated. This yields a full PDE system for a compressible two-phase fluid with mass transfer of the gaseous species. Then the passage to an incompressible solvent in the liquid phase is discussed, where a carefully chosen equation of state for the liquid mixture pressure allows for a limit in which the solvent density is constant. We finally provide a simplification of the PDE system in case of a dilute solution.

9. On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary

   Miroslav Bulíček, Josef Málek

   Abstract

   In the analysis of weak solutions relevant to evolutionary flows of incompressible fluids with non-constant viscosity or with non-linear constitutive equation, it is in general an open question whether a globally integrable pressure exists if the flows are subject to no-slip boundary conditions. Here we overcome this deficiency by considering threshold boundary conditions stating that the fluid adheres to the boundary until certain critical value for the wall shear stress is reached. Once the wall shear stress exceeds this critical value, the fluid slips. The main ingredient in our approach is to look at this type of activated, stick-slip, boundary condition as an implicit constitutive equation on the boundary.

   We prove the long-time and large-data existence of weak solutions, with integrable pressure, to unsteady internal flows of Bingham and Navier-Stokes fluids subject to such threshold slip boundary conditions.

10. Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

    Robert Denk, Tim Seger

    Abstract

    Uniform a priori estimates for parameter-elliptic boundary value problems are well-known if the underlying basic space equals \(L^{p}(\Omega )\). However, much less is known for the \(W_{p}^{s}(\Omega )\)-realization, s > 0, of a parameter-elliptic boundary value problem. We discuss a priori estimates and the generation of analytic semigroups for these realizations in various cases. The Banach scale method can be applied for homogeneous boundary conditions if the right-hand side satisfies certain compatibility conditions, while for the general case parameter-dependent norms are used. In particular, we obtain a resolvent estimate for the general situation where no analytic semigroup is generated.

11. Blow-Up Criterion for 3D Navier-Stokes Equations and Landau-Lifshitz System in a Bounded Domain

    Jishan Fan, Tohru Ozawa

    Abstract

    In this paper we prove a blow-up criterion for the 3D Navier-Stokes equations in a bounded domain in terms of a BMO norm of vorticity. We will also prove a regularity criterion for the Landau-Lifshitz system in a bounded domain.

12. Local Regularity Results for the Instationary Navier-Stokes Equations Based on Besov Space Type Criteria

    Reinhard Farwig

    Abstract

    Consider a weak instationary solution u of the Navier-Stokes equations in a domain \(\Omega \subset \mathbb{R}^{3}\), i.e.,

    $$u \in L^{\infty }\big(0,T;L^{2}(\Omega )\big) \cap L^{2}\big(0,T;W_{0}^{1,2}(\Omega )\big)$$

    and u solves the Navier-Stokes system in the sense of distributions. It is a famous open problem whether weak solutions are unique and smooth. A main step in the analysis of this problem is to show that the given weak solution is a strong one in the sense of J. Serrin, i.e., \(u \in L^{s}\big(0,T;L^{q}(\Omega )\big)\) where s > 2, q > 3 and \(\frac{2} {s} + \frac{3} {q} = 1\). In this review we report on recent results on this problem, considering first of all optimal initial values u(0) to yield a local in time strong solution, then criteria to prove regularity locally on subintervals of [0, T). Special emphasis is put on results for smooth bounded and also general unbounded domains. Most criteria are based on conditions of Besov space type.

13. On Global Well/Ill-Posedness of the Euler-Poisson System

    Eduard Feireisl

    Abstract

    We discuss the problem of well-posedness of the Euler-Poisson system arising, for example, in the theory of semi-conductors, models of plasma and gaseous stars in astrophysics. We introduce the concept of dissipative weak solution satisfying, in addition to the standard system of integral identities replacing the original system of partial differential equations, the balance of total energy, together with the associated relative entropy inequality. We show that strong solutions are unique in the class of dissipative solutions (weak-strong uniqueness). Finally, we use the method of convex integration to show that the Euler-Poisson system may admit even infinitely many weak dissipative solutions emanating from the same initial data.

14. On the Motion of a Liquid-Filled Rigid Body Subject to a Time-Periodic Torque

    Giovanni P. Galdi, Giusy Mazzone, Mahdi Mohebbi

    Abstract

    In this paper we investigate the existence of time-periodic motions of a system constituted by a rigid body with an interior cavity completely filled with a viscous liquid, and subject to a time-periodic external torque acting on the rigid body. We then show that the system of equations governing the motion of the coupled system liquid-filled rigid body, has at least one corresponding time-periodic weak solution. Furthermore if the size of the torque is below a certain constant, the weak solution is in fact strong.

15. Seeking a Proof of Xie’s Inequality: On the Conjecture That $$\mu _{m} \rightarrow \infty $$

    John G. Heywood

    Abstract

    I pursue an argument of Wenzheng Xie, as furthered in several of my papers, to prove a particular point-wise bound for solutions of the three-dimensional steady Stokes problem. If proven, it will provide the basis for an existence and regularity theory for the non-stationary Navier-Stokes equations, free of assumptions about the regularity of the boundary of the flow region. It will be valid for flow in an arbitrary open set.

    In his doctoral thesis, Xie proved an analogous bound for solutions of the Poisson problem for the Laplacian, considering it as a model problem. His proof carries over to the Stokes problem except at one point where the maximum principle is invoked. Subsequently, I’ve proposed a variant of Xie’s argument that circumvents the maximum principle, but requires instead a proof that a certain sequence of functions introduced in Xie’s argument tends to become singular. I’ve expressed this as a further conjecture, which is studied here for both the Stokes problem and for the Poisson problem, the latter being considered as a model problem.

16. Bounded Analyticity of the Stokes Semigroup on Spaces of Bounded Functions

    Matthias Hieber, Paolo Maremonti

    Abstract

    Let \(\Omega \subset \mathbb{R}^{n}\), n ≥ 3, be an exterior domain with smooth boundary. It is shown that the Stokes semigroup on \(L_{\sigma }^{\infty }(\Omega )\) is a bounded analytic semigroup on this space.

17. On the Weak Solution of the Fluid-Structure Interaction Problem for Shear-Dependent Fluids

    Anna Hundertmark, Mária Lukáčová-Medviďová, Šárka Nečasová

    Abstract

    In this paper the coupled fluid-structure interaction problem for incompressible non-Newtonian shear-dependent fluid flow in two-dimensional time-dependent domain is studied. One part of the domain boundary consists of an elastic wall. Its temporal evolution is governed by the generalized string equation with action of the fluid forces by means of the Neumann type boundary condition. The aim of this work is to present the limiting process for the auxiliary \((\kappa,\varepsilon,k)\)-problem. The weak solution of this auxiliary problem has been studied in our recent work (Hundertmark-Zaušková, Lukáčová-Medvid​’ová, Nečasová, On the existence of weak solution to the coupled fluid-structure interaction problem for non-Newtonian shear-dependent fluid, J. Math. Soc. Japan (in press)).

18. Stability of Time Periodic Solutions for the Rotating Navier-Stokes Equations

    Tsukasa Iwabuchi, Alex Mahalov, Ryo Takada

    Abstract

    We consider the stability problem of time periodic solutions for the rotating Navier-Stokes equations. For the non-rotating case, it is known that time periodic solutions to the original Navier-Stokes equations are asymptotically stable under the smallness assumptions both on the time periodic solutions and on the initial disturbances. We shall treat the high-rotating cases, and prove the asymptotic stability of large time periodic solutions for large initial perturbations.

19. Weighted L p − L q Estimates of Stokes Semigroup in Half-Space and Its Application to the Navier-Stokes Equations

    Takayuki Kobayashi, Takayuki Kubo

    Abstract

    We consider the Navier-Stokes equations in half-space and in L ^p space with Muckenhoupt weight and show the \(L^{p} - L^{q}\) estimates of Stokes semigroup with \(\langle x'\rangle ^{s_{1}}\langle x_{n}\rangle ^{s_{n}}\) type weight. Finally as the application of the weighted \(L^{p} - L^{q}\) estimates, we shall obtain the weighted asymptotic behavior of the solution to the Navier-Stokes equations.

20. On Vorticity Formulation for Viscous Incompressible Flows in R + 3

    Humiya Kosaka, Yasunori Maekawa

    Abstract

    In this paper we study the vorticity equations for viscous incompressible flows in the half space under the no-slip boundary condition on the velocity field. In particular, the boundary condition for the vorticity field is presented explicitly, and the solution formula for the linearized problem is obtained.

21. A Weak Solution to the Navier–Stokes System with Navier’s Boundary Condition in a Time-Varying Domain

    Jiří Neustupa, Patrick Penel

    Abstract

    We assume that \(\Omega ^{t}\) (for t ∈ [0, T]) is a time-varying domain in \(\mathbb{R}^{3}\). Particularly, \(\Omega ^{t}\) can be a region around colliding bodies. Under certain conditions on \(\Omega ^{t}\) and the way it varies, we prove the weak solvability of the Navier–Stokes system with Navier’s slip boundary condition in \(Q_{(0,T)}:=\{ (\mathbf{x},t);\ 0 < t < T,\ \mathbf{x} \in \Omega ^{t}\}\).

22. Effects of Fluid-Boundary Interaction on the Stability of Boundary Layers in Plasma Physics

    Masashi Ohnawa

    Abstract

    The present article addresses the asymptotic stability of stationary solutions to a system of the Euler-Poisson equations with an interaction between fluids and boundaries. The system describes motions of positive ions in a boundary layer called a sheath in a plasma flow, formed under the Bohm criterion, which requires the outflow velocity to exceed the ion acoustic velocity. In this model, charged particles accumulate as the plasma arrive at the boundary, which, at the same time affects fluid flows by changing the electric field over the entire domain through the change in the boundary condition. We show that the boundary layer is asymptotically stable under this fluid-boundary interactive setting.

23. On Incompressible Two-Phase Flows with Phase Transitions and Variable Surface Tension

    Jan Prüss, Senjo Shimizu, Gieri Simonett, Mathias Wilke

    Abstract

    Our study of the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics (Prüss et al., Evol Equ Control Theory 1:171–194, 2012; Prüss and Shimizu, J Evol Equ 12:917–941, 2012; Prüss et al., Commun Part Differ Equ 39:1236–1283, 2014; see also Prüss et al., Interfaces Free Bound 15:405–428, 2013) is extended to the case of temperature-dependent surface tension. We prove well-posedness in an L _p -setting, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exists globally, and if its limit set contains a stable equilibrium it converges to this equilibrium in the natural state manifold for the problem as time goes to infinity.

24. On the Nash-Moser Iteration Technique

    Paolo Secchi

    Abstract

    The aim of this work is to provide a brief presentation of the Nash-Moser iteration technique for the resolution of nonlinear equations, where the linearized equations admit estimates with a loss of regularity with respect to the given data.

25. Rate of Convergence to the Stationary Solution of the Navier-Stokes Exterior Problem

    Masao Yamazaki

    Abstract

    This paper is concerned with the nonstationary Navier-Stokes equation in two-dimensional exterior domains with stationary external forces, and provides the rate of convergence of solutions to the stationary solution under the smallness condition of the stationary solution.