Submit to Mathematics Review for Mathematics Propose a Special Issue

Journal Menu

Journal Browser

Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics II

Print Special Issue Flyer
Special Issue Editors
Special Issue Information
Keywords
Published Papers

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 17017

Share This Special Issue

Special Issue Editor

Prof. Dr. Yoshihiro Shibata

E-Mail Website
Guest Editor

Department of Mathematics, Waseda Univeristy, Tokyo 169-8555, Japan
Interests: R-boundedness; maximal regularity; mathematical fluid mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The unique existence of strong solutions appearing in mathematical fluid dynamics, such as Navier–Stokes equations, MHD, etc., is one of the main subjects in the study of nonlinear partial differential equations. Moreover, the maximal regularity and stability estimates for linearized equations play the most important role in analysis today. Due to the many recent developments in the area, I would like to organize a Special Issue contributing to this mathematical investigation.

Prof. Dr. Yoshihiro Shibata
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Maximal regularity
Navier–Stokes equations
Free boundary problems
Moving boundary problem
Stationary problem
Time periodic solutions
Stability estimates

Published Papers (7 papers)

Download All Papers

Order results

Result details

Show export options Show export options

Select all

Export citation of selected articles as:

Research

41 pages, 501 KiB

Open AccessArticle

Local Solvability for a Compressible Fluid Model of Korteweg Type on General Domains

by Suma Inna and Hirokazu Saito

Mathematics 2023, 11(10), 2368; https://doi.org/10.3390/math11102368 - 19 May 2023

Cited by 1 | Viewed by 820

Abstract

In this paper, we consider a compressible fluid model of the Korteweg type on general domains in the N-dimensional Euclidean space for

N \geq 2

. The Korteweg-type model is employed to describe fluid capillarity effects or liquid–vapor two-phase flows with phase [...] Read more.

In this paper, we consider a compressible fluid model of the Korteweg type on general domains in the N-dimensional Euclidean space for

N \geq 2

. The Korteweg-type model is employed to describe fluid capillarity effects or liquid–vapor two-phase flows with phase transition as a diffuse interface model. In the Korteweg-type model, the stress tensor is given by the sum of the standard viscous stress tensor and the so-called Korteweg stress tensor, including higher order derivatives of the fluid density. The local existence of strong solutions is proved in an

L_{p}

-in-time and

L_{q}

-in-space setting,

p \in (1, \infty)

and

q \in (N, \infty)

, with additional regularity of the initial density on the basis of maximal regularity for the linearized system. Full article

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

26 pages, 392 KiB

Open AccessFeature PaperArticle

Global Well-Posedness for the Compressible Nematic Liquid Crystal Flows

by Miho Murata

Mathematics 2023, 11(1), 181; https://doi.org/10.3390/math11010181 - 29 Dec 2022

Viewed by 1153

Abstract

In this paper, we prove the unique existence of global strong solutions and decay estimates for the simplified Ericksen–Leslie system describing compressible nematic liquid crystal flows in

R^{N}

3 \leq N \leq 7

. Firstly, we rewrite the system in Lagrange [...] Read more.

In this paper, we prove the unique existence of global strong solutions and decay estimates for the simplified Ericksen–Leslie system describing compressible nematic liquid crystal flows in

R^{N}

3 \leq N \leq 7

. Firstly, we rewrite the system in Lagrange coordinates, and secondly, we prove the global well-posedness for the transformed system, which is the main task in this paper. The proof is based on the maximal

L_{p}

L_{q}

regularity and the

L_{p}

L_{q}

decay estimates to the linearized problem. Full article

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

25 pages, 388 KiB

Open AccessArticle

Global Dynamics of the Compressible Fluid Model of the Korteweg Type in Hybrid Besov Spaces

by Zihao Song and Jiang Xu

Mathematics 2023, 11(1), 174; https://doi.org/10.3390/math11010174 - 29 Dec 2022

Viewed by 894

Abstract

We are concerned with a system of equations governing the evolution of isothermal, viscous, and compressible fluids of the Korteweg type, which is used to describe a two-phase liquid–vapor mixture. It is found that there is a “regularity-gain" dissipative structure of linearized systems [...] Read more.

P^{'} (ρ^{*}) = 0

, in comparison with the classical compressible Navier–Stokes equations. First, we establish the global-in-time existence of strong solutions in hybrid Besov spaces by using Banach’s fixed point theorem. Furthermore, we prove that the global solutions with critical regularity are Gevrey analytic in fact. Secondly, based on Gevrey’s estimates, we obtain uniform bounds on the growth of the analyticity radius of solutions in negative Besov spaces, which lead to the optimal time-decay estimates of solutions and their derivatives of arbitrary order. Full article

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

17 pages, 384 KiB

Open AccessArticle

On the Regularity of Weak Solutions to Time-Periodic Navier–Stokes Equations in Exterior Domains

by Thomas Eiter

Mathematics 2023, 11(1), 141; https://doi.org/10.3390/math11010141 - 27 Dec 2022

Viewed by 996

Abstract

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities. Full article

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

28 pages, 452 KiB

Open AccessArticle

Hölder Space Theory for the Rotation Problem of a Two-Phase Drop

by Irina V. Denisova and Vsevolod A. Solonnikov

Mathematics 2022, 10(24), 4799; https://doi.org/10.3390/math10244799 - 16 Dec 2022

Cited by 1 | Viewed by 8689

Abstract

We investigate a uniformly rotating finite mass consisting of two immiscible, viscous, incompressible self-gravitating fluids which is governed by an interface problem for the Navier–Stokes system with mass forces and the gradient of the Newton potential on the right-hand sides. The interface between the liquids is assumed to be closed. Surface tension acts on the interface and on the exterior free boundary. A study of this problem is performed in the Hölder spaces of functions. The global unique solvability of the problem is obtained under the smallness of the initial data, external forces and rotation speed, and the proximity of the given initial surfaces to some axisymmetric equilibrium figures. It is proved that if the second variation of the energy functional is positive and mass forces decrease exponentially, then small perturbations of the axisymmetric figures of equilibrium tend exponentially to zero as the time

t \to \infty

, and the motion of liquid mass passes into the rotation of the two-phase drop as a solid body. Full article

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

► Show Figures

Figure 1

45 pages, 523 KiB

Open AccessArticle

Local Well-Posedness for the Magnetohydrodynamics in the Different Two Liquids Case

by Elena Frolova and Yoshihiro Shibata

Mathematics 2022, 10(24), 4751; https://doi.org/10.3390/math10244751 - 14 Dec 2022

Cited by 1 | Viewed by 1995

Abstract

We consider the free boundary problem of MHD in the multi-dimensional case. This problem describes the motion of two incompressible fluids separated by a closed interface under the action of a magnetic field. This problem is overdetermined, and we find an equivalent system [...] Read more.

L_{p}

L_{q}

maximal regularity class, where

1 < p, q < \infty

and

2 / p + N / q < 1

. As a result, the original two-phase problem for the MHD is solvable locally in time. Full article

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

37 pages, 479 KiB

Open AccessArticle

On the Global Well-Posedness and Decay of a Free Boundary Problem of the Navier–Stokes Equation in Unbounded Domains

by Kenta Oishi and Yoshihiro Shibata

Mathematics 2022, 10(5), 774; https://doi.org/10.3390/math10050774 - 28 Feb 2022

Cited by 1 | Viewed by 1455

Abstract

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in

L_{p}

in time and

L_{q}

in space framework in a uniformly

H_{\infty}^{2}

[...] Read more.

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in

L_{p}

in time and

L_{q}

in space framework in a uniformly

H_{\infty}^{2}

domain

Ω \subset R^{N}

for

N \geq 4

. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the

L_{q}

L_{r}

estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata. The restriction

N \geq 4

is required to deduce an estimate for the nonlinear term

G (u)

arising from

{div}_{} v = 0

. However, we establish the results in the half space

R_{+}^{N}

for

N \geq 3

by reducing the linearized problem to the problem with

G = 0

, where

G

is the right member corresponding to

G (u)

. Full article

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

Show export options Show export options

Select all

Export citation of selected articles as:

Displaying articles 1-7

Journal Menu

Journal Browser

Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics II

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Published Papers (7 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI