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Symmetry
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Advances in Mathematical Models and Partial Differential Equations: 2nd Edition
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Advances in Mathematical Models and Partial Differential Equations: 2nd Edition

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 2555

Special Issue Editors

Prof. Dr. Yunhu Wang

E-Mail Website
Guest Editor
College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China
Interests: integral systems
Dr. Manwai Yuen

E-Mail Website
Guest Editor
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
Interests: partial differential equations; symmetry reduction; blowup; euler-poisson equations; euler equations with or without coriolis force; camassa-holm equations; navier-stokes equations; Magnetohydrodynamics (MHD); analytical and exact solutions; mathematical methods in fluids; classical cosmology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the study of partial differential equations (PDE), “blow-up” or “singularity” means the breakdown of a system within a finite time. The singularity formation in nonlinear physical systems has attracted the attention of many physics and mathematics researchers because of its physical significance and mathematical challenge. In this regard, a PDE system's lifespan is the maximum time before the solutions exist and are sufficiently smooth.

In the study of PDE, symmetry assumptions or reductions are expected to facilitate the study of the lifespan of the nonlinear partial differential systems. In other words, symmetry is especially useful to analyze simpler cases of some complex systems.

In this Special Issue, we expect that a theoretical or numerical study of the lifespan of nonlinear PDE, can be developed. To contribute to this Special Issue, we expect that the theoretical analysis can establish a sufficient condition on initial data that guarantees that the lifespan of the systems is finite. For the numerical study of the lifespan problem, the maximal existence time must be estimated with significant improvement.

Prof. Dr. Yunhu Wang
Dr. Manwai Yuen
Guest Editors

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. Symmetry is an international peer-reviewed open access monthly 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 2400 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

  • partial differential equation
  • mathematical method

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Published Papers (6 papers)

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Research

19 pages, 284 KiB  
Article
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
Viewed by 196
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 Xa,γρ(R3), 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 Xa,γρ(R3), where ρ∈[−1,0),a>0, and γ∈(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 Xρ(R3), 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)
18 pages, 2823 KiB  
Article
Quasi-Periodic Dynamics and Wave Solutions of the Ivancevic Option Pricing Model Using Multi-Solution Techniques
by Sadia Yasin, Fehaid Salem Alshammari, Asif Khan and Beenish
Symmetry 2025, 17(7), 1137; https://doi.org/10.3390/sym17071137 - 16 Jul 2025
Viewed by 169
Abstract
In this research paper, we study symmetry groups, soliton solutions, and the dynamical behavior of the Ivancevic Option Pricing Model (IOPM). First, we find the Lie symmetries of the considered model; next, we use them to determine the corresponding symmetry groups. Then, we [...] Read more.
In this research paper, we study symmetry groups, soliton solutions, and the dynamical behavior of the Ivancevic Option Pricing Model (IOPM). First, we find the Lie symmetries of the considered model; next, we use them to determine the corresponding symmetry groups. Then, we attempt to solve IOPM by means of two methods. We provide some wave solutions and give further details of the solution using 2D and 3D graphs. These results are interpreted as important clarifications in financial mathematics and deepen our understanding of the dynamics involved during the pricing of options. Secondly, the quasi-periodic behavior of the two-dimensional dynamical system and its perturbed system are plotted using Python software (Python 3.13.5 version). Various frequencies and amplitudes are considered to confirm the quasi-periodic behavior via the Lyapunov exponent, bifurcation diagram, and multistability analysis. These findings are particularly in consonance with current research that investigates IOPM as a nonlinear wave alternate for normal models and the importance of graphical representations in the understanding of financial derivative dynamics. We, therefore, hope to fill in the gaps in the literature that currently exist about the use of multi-solution methods and their effects on financial modeling through the employment of sophisticated graphical techniques. This will be helpful in discussing matters in the field of financial mathematics and open up new directions of investigation. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
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18 pages, 312 KiB  
Article
Lipschitz and Second-Order Regularities for Non-Homogeneous Degenerate Nonlinear Parabolic Equations in the Heisenberg Group
by Huiying Wang, Chengwei Yu, Zhiqiang Zhang and Yue Zeng
Symmetry 2025, 17(5), 799; https://doi.org/10.3390/sym17050799 - 21 May 2025
Viewed by 325
Abstract
In the Heisenberg group Hn, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form [...] Read more.
In the Heisenberg group Hn, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ∂tu−∑i=12nXiAi(Xu)=K(x,t,u,Xu), where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For 2≤p≤4, we establish the local Lipschitz regularity (u∈Cloc0,1), with the horizontal gradient satisfying Xu∈Lloc∞; (ii) For 2≤p<3, we establish the local second-order horizontal Sobolev regularity (u∈HWloc2,2), with the second-order horizontal derivative satisfying XXu∈Lloc2. These results solve an open problem proposed by Capogna et al. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
29 pages, 2593 KiB  
Article
Symmetry and Time-Delay-Driven Dynamics of Rumor Dissemination
by Cunlin Li, Zhuanting Ma, Lufeng Yang and Tajul Ariffin Masron
Symmetry 2025, 17(5), 788; https://doi.org/10.3390/sym17050788 - 19 May 2025
Viewed by 372
Abstract
The dissemination of rumors can lead to significant economic damage and pose a grave threat to social harmony and the stability of people’s livelihoods. Consequently, curbing the dissemination of rumors is of paramount importance. The model in the text assumes that the population [...] Read more.
The dissemination of rumors can lead to significant economic damage and pose a grave threat to social harmony and the stability of people’s livelihoods. Consequently, curbing the dissemination of rumors is of paramount importance. The model in the text assumes that the population is homogeneous in terms of transmission behavior. This homogeneity is essentially a manifestation of translational symmetry. This paper undertakes a thorough examination of the impact of time delay on the dissemination of rumors within social networking services. We have developed a model for rumor dissemination, establishing the positivity and boundedness of its solutions, and identified the existence of an equilibrium point. The study further involved determining the critical threshold of the proposed model, accompanied by a comprehensive examination of its Hopf bifurcation characteristics. In the expression of the threshold R0, the parameters appear in a symmetric form, reflecting the balance between dissemination and suppression mechanisms. Furthermore, detailed investigations were carried out to assess both the localized and global stability properties of the system’s equilibrium states. In stability analysis, the symmetry in the distribution of characteristic equation roots determines the system’s dynamic behavior. Through numerical simulations, we analyzed the potential impacts and theoretically examined the factors influencing rumor dissemination, thereby validating our theoretical analysis. An optimal control strategy was formulated, and three control variables were incorporated to describe the strategy. The optimization framework incorporates a specifically designed cost function that simultaneously accounts for infection reduction and resource allocation efficiency in control strategy implementation. The optimal control strategy proposed in the study involves a comparison between symmetric and asymmetric interventions. Symmetric control measures may prove inefficient, whereas asymmetric control demonstrates higher efficacy—highlighting a trade-off in symmetry considerations for optimization problems. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
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19 pages, 3215 KiB  
Article
Characteristic Analysis of Local Wave Solutions for the (21)-Dimensional Asymmetric Nizhnik–Novikov–Veselov Equation
by Jingyi Chu, Yaqing Liu, Huining Wu and Manwai Yuen
Symmetry 2025, 17(4), 514; https://doi.org/10.3390/sym17040514 - 28 Mar 2025
Viewed by 258
Abstract
This study investigates the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation, a significant model in nonlinear science, using the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Despite the extensive research on the ANNV equation, a comprehensive exploration of its solutions using the KP hierarchy reduction method is [...] Read more.
This study investigates the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation, a significant model in nonlinear science, using the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Despite the extensive research on the ANNV equation, a comprehensive exploration of its solutions using the KP hierarchy reduction method is lacking. This gap is addressed by identifying constraint conditions that transform a specific KP hierarchy equation into the ANNV equation, thereby enabling the derivation of its Gram determinant solutions. By selecting appropriate τ functions, we obtain breather solutions and analyze their dynamic behavior during wave oscillations. Additionally, lump solutions are derived through long-wave limit analysis, revealing their unique characteristics. This study further explores hybrid solutions that combine breathers and lumps, providing new insights to the interaction between these localized wave phenomena. Our findings enhance the understanding of the ANNV equation’s dynamics and contribute to the broader field of nonlinear wave theory. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
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19 pages, 292 KiB  
Article
On the Cauchy Problem for the Vlasov-Maxwell-Fokker-Planck System in Low Regularity Space
by Yingzhe Fan and Lihua Tan
Symmetry 2025, 17(1), 100; https://doi.org/10.3390/sym17010100 - 10 Jan 2025
Viewed by 644
Abstract
In this study, we investigate the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space. We establish the existence of global mild solutions to the system by employing the energy method, provided that the perturbative initial data is [...] Read more.
In this study, we investigate the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space. We establish the existence of global mild solutions to the system by employing the energy method, provided that the perturbative initial data is sufficiently small. Moreover, despite the absence of zeroth-order dissipation for the magnetic field, we are able to derive exponential decay estimates for solutions in higher-order regularity space. This is achieved by leveraging the higher-order dissipation properties of the magnetic field, which are deduced from the Maxwell equation. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
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