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22 pages, 373 KB

Open AccessArticle

Fractional Viscous–Resistive Magnetohydrodynamics at Critical Scales: Global Solutions and Gevrey Regularity

by Siyi Xie, Chengzhou Wei and Muhammad Zainul Abidin

Axioms 2026, 15(5), 372; https://doi.org/10.3390/axioms15050372 - 16 May 2026

Viewed by 110

We study the incompressible fractional viscous–resistive magnetohydrodynamic system on

R^{n}

with fractional diffusion

{(- Δ)}^{α}

, where

α \in (1 / 2, 1]

, and with positive viscosity and resistivity coefficients

μ, ν > 0

[...] Read more.

We study the incompressible fractional viscous–resistive magnetohydrodynamic system on

R^{n}

with fractional diffusion

{(- Δ)}^{α}

, where

α \in (1 / 2, 1]

, and with positive viscosity and resistivity coefficients

μ, ν > 0

. The problem is treated at the scale-invariant regularity

s_{c} = \frac{n}{p} + 1 - 2 α .

For small divergence-free initial data in the critical Triebel–Lizorkin–Lorentz space

{\dot{F}}_{p, r}^{s_{c}, q}

, we construct a unique global mild solution. The main contribution is the use of the single-norm time–frequency space

m_{m}^{'} {\dot{F}}_{p, r}^{s_{c}, q}

, built on Meyer wavelets and the parabolic gauge

t 2^{2 α j}

. This space keeps the critical spatial size, the short-time behavior, and the high-frequency decay in one norm. By using a Gevrey-weighted Duhamel formulation, we prove boundedness of the corresponding fractional heat propagators and establish the bilinear paraproduct estimate required for the fixed-point argument. Consequently,

e^{{(t {(- Δ)}^{α})}^{γ}} (u, b) \in {(m_{m}^{'} {\dot{F}}_{p, r}^{s_{c}, q})}^{2 n}

for some

γ > 0

depending on the parameters. This gives a Gevrey-type spatial smoothing effect, which is stronger than ordinary analyticity in the adopted scale. The restriction

α > \frac{1}{2}

enters through the factor

2^{j (1 - 2 α)}

, which supplies the high-frequency gain needed to close the critical bilinear estimates; in this sense it is sharp for the present method. The classical viscous–resistive case is recovered when

α = 1

. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

19 pages, 284 KB

Open AccessArticle

Local and Global Solutions of the 3D-NSE in Homogeneous Lei–Lin–Gevrey Spaces

by Lotfi Jlali

Symmetry 2025, 17(7), 1138; https://doi.org/10.3390/sym17071138 - 16 Jul 2025

Cited by 1 | Viewed by 717

Abstract

This paper investigates the existence and uniqueness of local and global solutions to the incompressible three-dimensional Navier–Stokes equations within the framework of homogeneous Lei–Lin–Gevrey spaces

X_{a, γ}^{ρ} (R^{3})

, where [...] Read more.

This paper investigates the existence and uniqueness of local and global solutions to the incompressible three-dimensional Navier–Stokes equations within the framework of homogeneous Lei–Lin–Gevrey spaces

X_{a, γ}^{ρ} (R^{3})

, where

ρ \in [- 1, 0), a > 0

, and

γ \in (0, 1)

. These function spaces combine the critical scaling structure of the Lei–Lin spaces with the exponential regularity of Gevrey classes, thereby enabling a refined treatment of analytic regularity and frequency localization. The main results are obtained under the assumption of small initial data in the critical Lei–Lin space

X^{ρ} (R^{3})

, extending previous works and improving regularity thresholds. In particular, we establish that for suitable initial data, the Navier–Stokes system admits unique solutions globally in time. The influence of the Gevrey parameter

γ

on the high-frequency behavior of solutions is also discussed. This work contributes to a deeper understanding of regularity and decay properties in critical and supercritical regimes. Full article

(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)

28 pages, 400 KB

Open AccessArticle

Global Analysis of Compressible Navier–Stokes–Korteweg Equations: Well-Posedness and Gevrey Analyticity

by Jianzhong Zhang, Weixuan Shi and Minggang Han

Axioms 2025, 14(6), 411; https://doi.org/10.3390/axioms14060411 - 28 May 2025

Viewed by 1103

Abstract

This paper investigates the Cauchy problem for the full compressible Navier–Stokes–Korteweg equations, which model fluid dynamics with capillary properties in

R^{d} (d \geq 3)

. And the global well-posedness and Gevrey analytic of strong solutions for the system are established [...] Read more.

This paper investigates the Cauchy problem for the full compressible Navier–Stokes–Korteweg equations, which model fluid dynamics with capillary properties in

R^{d} (d \geq 3)

. And the global well-posedness and Gevrey analytic of strong solutions for the system are established in the

L^{2} - L^{p}

type critical hybrid Besov space with

2 \leq p \leq \frac{2 d}{d - 2}

and

p < d

. Full article

15 pages, 296 KB

Open AccessArticle

New Results on Gevrey Well Posedness for the Schrödinger–Korteweg–De Vries System

by Feriel Boudersa, Abdelaziz Mennouni and Ravi P. Agarwal

Math. Comput. Appl. 2025, 30(3), 52; https://doi.org/10.3390/mca30030052 - 7 May 2025

Cited by 1 | Viewed by 851

Abstract

In this work, we prove that the initial value problem for the Schrödinger–Korteweg–de Vries (SKdV) system is locally well posed in Gevrey spaces for

s > - \frac{3}{4}

and

k \geq 0

. This advancement extends recent findings regarding the well posedness [...] Read more.

In this work, we prove that the initial value problem for the Schrödinger–Korteweg–de Vries (SKdV) system is locally well posed in Gevrey spaces for

s > - \frac{3}{4}

and

k \geq 0

. This advancement extends recent findings regarding the well posedness of this model within Sobolev spaces and investigates the regularity properties of its solutions. Full article

24 pages, 374 KB

Open AccessArticle

An Introduction to Extended Gevrey Regularity

by Nenad Teofanov, Filip Tomić and Milica Žigić

Axioms 2024, 13(6), 352; https://doi.org/10.3390/axioms13060352 - 24 May 2024

Cited by 3 | Viewed by 2311

Abstract

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in [...] Read more.

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in Gevrey settings. In this paper, we consider a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview of extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultra distributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic. Full article

(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)

25 pages, 388 KB

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 1908

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.

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 in case of zero sound speed

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)

18 pages, 356 KB

Open AccessArticle

Continuity and Analyticity for the Generalized Benjamin–Ono Equation

by Xiaolin Pan, Bin Wang and Rong Chen

Symmetry 2021, 13(12), 2435; https://doi.org/10.3390/sym13122435 - 16 Dec 2021

Viewed by 2735

Abstract

This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such [...] Read more.

This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces

H^{s} (R)

with

s > 3 / 2

. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in

H^{r}

-topology for all

0 \leq r < s

with exponent

α

depending on s and r. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map. Full article

(This article belongs to the Topic Dynamical Systems: Theory and Applications)

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16 pages, 798 KB

Open AccessArticle

Well-Posedness and Time Regularity for a System of Modified Korteweg-de Vries-Type Equations in Analytic Gevrey Spaces

by Aissa Boukarou, Kaddour Guerbati, Khaled Zennir, Sultan Alodhaibi and Salem Alkhalaf

Mathematics 2020, 8(5), 809; https://doi.org/10.3390/math8050809 - 16 May 2020

Cited by 13 | Viewed by 2927

Abstract

Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value [...] Read more.

Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value problem associated with a coupled system consisting of modified Korteweg-de Vries equations for given data. Furthermore, we prove that the unique solution belongs to Gevrey space

G^{σ} \times G^{σ}

in x and

G^{3 σ} \times G^{3 σ}

in t. This article is a continuation of recent studies reflected. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications)

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