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 July 2025 | Viewed by 1102

Special Issue Editors


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Guest Editor
College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China
Interests: integral systems

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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
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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

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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.

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Keywords

  • partial differential equation
  • mathematical method

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Published Papers (2 papers)

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Research

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 153
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
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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 513
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
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