Research

28 pages, 396 KiB

Open AccessArticle

Weak Solutions and Their Kinetic Energy Regarding Time-Periodic Navier–Stokes Equations in Three Dimensional Whole-Space

by Mads Kyed

Mathematics 2021, 9(13), 1528; https://doi.org/10.3390/math9131528 - 29 Jun 2021

Viewed by 1079

Abstract

The existence of weak time-periodic solutions to Navier–Stokes equations in three dimensional whole-space with time-periodic forcing terms are established. The solutions are constructed in such a way that the structural properties of their kinetic energy are obtained. No restrictions on either the size [...] Read more.

The existence of weak time-periodic solutions to Navier–Stokes equations in three dimensional whole-space with time-periodic forcing terms are established. The solutions are constructed in such a way that the structural properties of their kinetic energy are obtained. No restrictions on either the size or structure of the external force are required. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

14 pages, 307 KiB

Open AccessArticle

Navier–Stokes Cauchy Problem with |v₀(x)|² Lying in the Kato Class K₃

by Francesca Crispo and Paolo Maremonti

Mathematics 2021, 9(11), 1167; https://doi.org/10.3390/math9111167 - 22 May 2021

Cited by 1 | Viewed by 1261

Abstract

We investigate the 3D Navier–Stokes Cauchy problem. We assume the initial datum

v_{0}

is weakly divergence free,

sup_{R^{3}}

\int_{R^{3}} \frac{| v_{0} {(y) |}^{2}}{| x - y |} d y < \infty

and [...] Read more.

We investigate the 3D Navier–Stokes Cauchy problem. We assume the initial datum

v_{0}

is weakly divergence free,

sup_{R^{3}}

\int_{R^{3}} \frac{| v_{0} {(y) |}^{2}}{| x - y |} d y < \infty

and

| v_{0} {(y) |}^{2} \in K_{3}

, where

K_{3}

denotes the Kato class. The existence is local for arbitrary data and global if

sup_{R^{3}}

\int_{R^{3}} \frac{| v_{0} {(y) |}^{2}}{| x - y |} d y

is small. Regularity and uniqueness also hold. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

31 pages, 409 KiB

Open AccessArticle

Weak Solutions to a Fluid-Structure Interaction Model of a Viscous Fluid with an Elastic Plate under Coulomb Friction Coupling

by Reinhard Farwig and Andreas Schmidt

Mathematics 2021, 9(9), 1026; https://doi.org/10.3390/math9091026 - 01 May 2021

Cited by 2 | Viewed by 1039

Abstract

We consider the behavior of a viscous fluid within a container that has an elastic upper, free boundary. The movement of the upper boundary is described by a combination of a plate equation and a boundary condition of friction type that quantifies the [...] Read more.

We consider the behavior of a viscous fluid within a container that has an elastic upper, free boundary. The movement of the upper boundary is described by a combination of a plate equation and a boundary condition of friction type that quantifies the elasticity of the boundary. We show the local existence of weak solutions to this coupled system in three dimensions, by applying the Galerkin method to a regularized version of the problem and using a fixed-point argument afterwards. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

19 pages, 373 KiB

Open AccessArticle

Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

by Toshiaki Hishida

Mathematics 2021, 9(8), 841; https://doi.org/10.3390/math9080841 - 12 Apr 2021

Viewed by 1082

Abstract

In this expository paper, we study

L^{q}

-

L^{r}

decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the [...] Read more.

In this expository paper, we study

L^{q}

-

L^{r}

decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space

R^{n}

together with the energy relation imply the desired estimates in exterior domains provided

n \geq 3

. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

43 pages, 714 KiB

Open AccessArticle

Time-Decay Estimates for Linearized Two-Phase Navier–Stokes Equations with Surface Tension and Gravity

by Hirokazu Saito

Mathematics 2021, 9(7), 761; https://doi.org/10.3390/math9070761 - 01 Apr 2021

Viewed by 1564

Abstract

The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane [...] Read more.

The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane

x_{N} = 0

in the N-dimensional Euclidean space,

N \geq 2

. It is well-known that the Rayleigh–Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of

L_{p}

-

L_{q}

type for the linearized equations. Our approach is based on solution formulas for a resolvent problem associated with the linearized equations. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

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34 pages, 458 KiB

Open AccessArticle

Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

by Yasunori Maekawa and Yoshihiro Ueda

Mathematics 2021, 9(7), 728; https://doi.org/10.3390/math9070728 - 28 Mar 2021

Viewed by 1583

Abstract

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, [...] Read more.

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

29 pages, 503 KiB

Open AccessArticle

On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid

by Giovanni P. Galdi

Mathematics 2021, 9(7), 715; https://doi.org/10.3390/math9070715 - 25 Mar 2021

Viewed by 1377

Abstract

We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere,

S

, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside

S

[...] Read more.

We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere,

S

, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside

S

), the main difficulty consists in finding the appropriate functional setting where general theory may apply. In this regard, we are able to show that the problem can be formulated as a suitable system of coupled operator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In such a way, we can use the theory recently introduced by the author and give sufficient conditions for time-periodic bifurcation to take place. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

36 pages, 469 KiB

Open AccessArticle

On the Spectral Properties for the Linearized Problem around Space-Time-Periodic States of the Compressible Navier–Stokes Equations

by Mohamad Nor Azlan, Shota Enomoto and Yoshiyuki Kagei

Mathematics 2021, 9(7), 696; https://doi.org/10.3390/math9070696 - 24 Mar 2021

Cited by 1 | Viewed by 1235

Abstract

This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of

R^{n}

(

n = 2, 3

), and the spectral properties of the linearized evolution operator is investigated. It is shown [...] Read more.

This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of

R^{n}

(

n = 2, 3

), and the spectral properties of the linearized evolution operator is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the asymptotic expansions of the Floquet exponents near the imaginary axis for the Bloch transformed linearized problem are obtained for small Bloch parameters, which would give the asymptotic leading part of the linearized solution operator as

t \to \infty

. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

20 pages, 341 KiB

Open AccessFeature PaperArticle

Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

by Takayuki Kobayashi, Masashi Misawa and Kazuyuki Tsuda

Mathematics 2021, 9(6), 683; https://doi.org/10.3390/math9060683 - 22 Mar 2021

Viewed by 1337

Abstract

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on

R^{2}

. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) [...] Read more.

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on

R^{2}

. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time

L^{2}

of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by

L^{2}

on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

44 pages, 561 KiB

Open AccessArticle

On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

by Takayuki Kubo and Yoshihiro Shibata

Mathematics 2021, 9(6), 621; https://doi.org/10.3390/math9060621 - 15 Mar 2021

Cited by 1 | Viewed by 1304

Abstract

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform

W_{q}^{2 - 1 / q}

domain in

R^{N}

( [...] Read more.

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform

W_{q}^{2 - 1 / q}

domain in

R^{N}

(

N \geq 2

). We prove the local in the time unique existence theorem for our problem in the

L_{p}

in time and

L_{q}

in space framework with

2 < p < \infty

and

N < q < \infty

under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal

L_{p}

-

L_{q}

regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an

R

-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with

R

-boundedness implies the generation of a continuous analytic semigroup and the maximal

L_{p}

-

L_{q}

regularity theorem. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

33 pages, 469 KiB

Open AccessArticle

Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

by Kenta Oishi and Yoshihiro Shibata

Mathematics 2021, 9(5), 461; https://doi.org/10.3390/math9050461 - 24 Feb 2021

Cited by 1 | Viewed by 1113

Abstract

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free [...] Read more.

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space

H_{p}^{1} ((0, T), H_{q}^{1}) \cap L_{p} ((0, T), H_{q}^{3})

for the velocity field and in an anisotropic space

H_{p}^{1} ((0, T), L_{q}) \cap L_{p} ((0, T), H_{q}^{2})

for the magnetic fields with

2 < p < \infty

,

N < q < \infty

and

2 / p + N / q < 1

. To prove our main result, we used the

L_{p}

-

L_{q}

maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

28 pages, 424 KiB

Open AccessArticle

Global Solvability of Compressible–Incompressible Two-Phase Flows with Phase Transitions in Bounded Domains

by Keiichi Watanabe

Mathematics 2021, 9(3), 258; https://doi.org/10.3390/math9030258 - 28 Jan 2021

Viewed by 1289

Abstract

Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains

Ω_{t +}, Ω_{t -} \subset R^{N}

,

N \geq 2

, where the domains are separated by a sharp compact interface [...] Read more.

Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains

Ω_{t +}, Ω_{t -} \subset R^{N}

,

N \geq 2

, where the domains are separated by a sharp compact interface

Γ_{t} \subset R^{N - 1}

. We prove a global in time unique existence theorem for such free boundary problem under the assumption that the initial data are sufficiently small and the initial domain of the incompressible fluid is close to a ball. In particular, we obtain the solution in the maximal

L_{p} - L_{q}

-regularity class with

2 < p < \infty

and

N < q < \infty

and exponential stability of the corresponding analytic semigroup on the infinite time interval. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics)

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