Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
4.6
2.2
Journals
Mathematics
Special Issues
Mathematical Fluid Dynamics: Theory, Analysis and Emerging Trends
share announcement

Mathematical Fluid Dynamics: Theory, Analysis and Emerging Trends

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 28 August 2026 | Viewed by 1030

Special Issue Editor

Prof. Dr. Shin'ichi Inage

E-Mail Website
Guest Editor
Department of Mechanical Engineering, Ishinomaki Senshu University, Fukuoka University, Fukuoka 986-8580, Japan
Interests: energy engineering; fluid numerical simulation; artificial intelligence

Special Issue Information

Dear Colleagues,

This Special Issue explores the mathematical essence of fluid motion through rigorous analysis, geometric reasoning, and modern data-driven methodologies. It seeks to illuminate how the foundational equations of continuum mechanics—most notably the Navier–Stokes and Euler equations—continue to pose profound challenges concerning existence, regularity, and singularity formation, while also inspiring new analytical and computational paradigms.

The Issue highlights advances in the mathematical theory of turbulence, the energy cascade across scales, and the analysis of free-boundary, multiphase, and magnetohydrodynamic (MHD) systems. Emphasis is placed on frameworks that integrate functional and harmonic analysis, stochastic formulations, and variational or geometric principles, thereby extending the reach of classical fluid theory. Moreover, the Issue welcomes mathematically grounded approaches to physics-informed and machine-learning-assisted modeling, which reconcile data adaptability with physical and analytical integrity. Collectively, these contributions aim to redefine the modern landscape of mathematical fluid dynamics, revealing how timeless equations give rise to new structures, symmetries, and pathways toward a deeper understanding of fluid motion in both nature and computation.

Prof. Dr. Shin'ichi Inage
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 250 words) can be sent to the Editorial Office for assessment.

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

  • Navier–Stokes equations
  • regularity and singularity
  • turbulence and energy cascade
  • free-boundary problems
  • multiphase and MHD flows
  • geometric and variational formulations
  • data-driven and physics-informed modeling

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • Reprint: MDPI Books provides the opportunity to republish successful Special Issues in book format, both online and in print.

Further information on MDPI's Special Issue policies can be found here.

Published Papers (2 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

17 pages, 284 KB  
Article
Linear Hamiltonian Vector Fields on Lie Groups
by Víctor Ayala and María Luisa Torreblanca Todco
Mathematics 2026, 14(6), 994; https://doi.org/10.3390/math14060994 - 14 Mar 2026
Viewed by 299
Abstract
Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type [...] Read more.
Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type geometric structure, a natural problem is to determine whether such linear dynamics admit a global variational realization, and how such realizations can be interpreted in terms of reduced models of fluid motion. In the even-dimensional case, where the Lie group carries a symplectic structure, we establish a complete global criterion for the existence of Hamiltonians generating linear symplectic vector fields. The problem reduces to a single global obstruction: the de Rham cohomology class of the 1-form ιXω. Thus, every linear symplectic vector field on a simply connected Lie group is globally Hamiltonian, and when the obstruction vanishes, we provide an explicit constructive procedure to recover the Hamiltonian. On the affine group Aff+(1), this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries. We then treat odd-dimensional Lie groups, where symplectic geometry is unavailable. Using contact geometry as the canonical replacement, we prove a Hamiltonian lifting theorem ensuring the existence and uniqueness of the associated dynamics. The Reeb vector field appears as a distinguished vertical direction resolving the ambiguities of degenerate Hamiltonian systems. On the Heisenberg group H3, this gives a fully explicit contact Hamiltonian model of an effective non-conservative fluid mode. Finally, we interpret symplectic and contact theories within a unified geometric framework and discuss their relevance to geometric formulations of ideal (symplectic) and effective (contact) fluid equations on Lie groups. Full article
(This article belongs to the Special Issue Mathematical Fluid Dynamics: Theory, Analysis and Emerging Trends)
44 pages, 940 KB  
Article
A Two-Level Relative-Entropy Theory for Isotropic Turbulence Spectra: Fokker–Planck Semigroup Irreversibility and WKB Selection of Dissipation Tails
by Shin-ichi Inage
Mathematics 2026, 14(4), 620; https://doi.org/10.3390/math14040620 - 10 Feb 2026
Viewed by 435
Abstract
We propose a two-level theory that connects Lin-equation-based dynamical coarse-graining of the turbulence cascade with an information-theoretic selection principle in logarithmic wavenumber space. This framework places the dissipation-range spectral shape on a verifiable logical basis rather than on ad hoc fitting. At the [...] Read more.
We propose a two-level theory that connects Lin-equation-based dynamical coarse-graining of the turbulence cascade with an information-theoretic selection principle in logarithmic wavenumber space. This framework places the dissipation-range spectral shape on a verifiable logical basis rather than on ad hoc fitting. At the first (dynamical) level, we formulate an autonomous conservative Fokker–Planck equation for the normalized density and probability current. Under sufficient boundary decay and a strictly positive effective diffusion, the sign-reversed Kullback–Leibler divergence is shown to be a Lyapunov functional, yielding a rigorous H-theorem and fixing the arrow of time in scale space. At the second (selection) level, the dissipation range is treated as a stationary boundary-value problem for an open system by introducing a killing term for an unnormalized scale density. A WKB (Liouville–Green) analysis restricts the admissible tail to a stretched-exponential form and links the tail exponent to the high-wavenumber scaling of the effective diffusion. The exponential prefactor is fixed by dissipation-rate consistency, and the remaining degree of freedom is determined by one-dimensional Kullback–Leibler minimization (Hyper-MaxEnt) against a globally constructed reference distribution. The resulting exponent range is validated against the high-resolution DNS spectra reported in the literature. Full article
(This article belongs to the Special Issue Mathematical Fluid Dynamics: Theory, Analysis and Emerging Trends)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-2
Back to TopTop