Analysis and applications of nonlinear partial differential equations in conservation laws and kinetic theories

非线性偏微分方程在守恒定律和动力学理论中的分析及应用

基本信息

  • 批准号:
    1108647
  • 负责人:
  • 金额:
    $ 10.46万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2011
  • 资助国家:
    美国
  • 起止时间:
    2011-09-15 至 2014-08-31
  • 项目状态:
    已结题

项目摘要

The principal investigator will study mathematical problems of certain nonlinear partial differential equations in fluid dynamics and the kinetic theory, such as the compressible relativistic Navier-Stokes equation, the magnetohydrodynamics (MHD) equations without the magnetic diffusivity, the compressible viscoelastic flows, and the Vlasov-Maxwell-Boltzmann equations. These equations model a variety of physical processes requiring significantly new mathematical approaches, and understanding of properties of these models is a fundamental challenge both from the mathematical and physical viewpoints. The research program consists of four parts:(1) The first part concerns the global existence of weak solutions of the relativistic Navier-Stokes equation and some qualitative properties of weak solutions. Compactness theory and the kinetic formulation will be explored.(2) The second part regards the global existence of strong solutions to the magnetohydrodynamics without magnetic diffusivity. The decay of the component of the magnetic field which is parallel to the equilibrium will be fully developed.(3) The third part considers the incompressible limit of the compressible viscoelastic fluids and the weak solutions to the two dimensional steady compressible viscoelastic fluids. The oscillation of the asymptotic solutions will be exploited.(4) The fourth part studies the Vlasov-Maxwell-Boltzmann equations.Topics that to be addressed include the global existence of the renormalized solution and its hydrodynamic limit.The mathematical problems to be investigated in the project arise in many scientific disciplines including plasma physics, elastodynamics, astrophysics, fluid dynamics, and the dynamics of biological and chemical reactions. The mathematical issues, such as global existence, incompressible limits, and hydrodynamic limits are of fundamental importance in biology, engineering, and physics. The primary goal of this research program is to provide mathematical verification of the fundamental models, to find a connection among these models, and to investigate analytical properties that will yield new insight into the mathematics of fluids, critical for plasma physics, elastodynamics, and other physical models, which could facilitate better understanding of the relevant physical phenomena.
主要研究者将研究流体动力学和动力学理论中某些非线性偏微分方程的数学问题,例如可压缩相对论Navier-Stokes方程,无磁扩散率的磁流体动力学(MHD)方程,可压缩粘弹性流动和Vlasov-Maxwell-Boltzmann方程。这些方程模型的各种物理过程需要显着新的数学方法,这些模型的属性的理解是一个根本的挑战,无论是从数学和物理的观点。研究内容包括四个部分:(1)第一部分研究了相对论性Navier-Stokes方程弱解的整体存在性及其定性性质。紧理论和动力学公式将被探讨。(2)第二部分考虑了无磁扩散系数的磁流体动力学问题强解的整体存在性。平行于平衡的磁场分量的衰减将充分发展。(3)第三部分研究了可压缩粘弹性流体的不可压缩极限和二维定常可压缩粘弹性流体的弱解。将利用渐近解的振荡性。(4)第四部分研究Vlasov-Maxwell-Boltzmann方程组,主要讨论重整化解的整体存在性及其流体力学极限,涉及到等离子体物理、弹性动力学、天体物理、流体动力学、生物和化学反应动力学等多个学科的数学问题。整体存在性、不可压缩极限和流体力学极限等数学问题在生物学、工程学和物理学中具有重要意义。该研究计划的主要目标是提供基本模型的数学验证,找到这些模型之间的联系,并研究分析特性,这些特性将产生对流体数学的新见解,对等离子体物理学,弹性动力学和其他物理模型至关重要,这可以促进更好地理解相关物理现象。

项目成果

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会议论文数量(0)
专利数量(0)

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Xianpeng Hu其他文献

Hausdorff Dimension of Concentration for Isentropic Compressible Navier–Stokes Equations
Global existence for the compressible viscoelastic flows with zero shear viscosity and general pressure law in three dimensions
三维零剪切粘度和一般压力定律的可压缩粘弹性流动的整体存在性
  • DOI:
    10.1016/j.jde.2025.02.069
  • 发表时间:
    2025-07-05
  • 期刊:
  • 影响因子:
    2.300
  • 作者:
    Xianpeng Hu;Song Meng;Ting Zhang
  • 通讯作者:
    Ting Zhang
Incompressible limit for compressible viscoelastic flows with large velocity
  • DOI:
    https://doi.org/10.1515/anona-2022-0324
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    4.2
  • 作者:
    Xianpeng Hu;Yaobin Ou;Dehua Wang;Lu Yang
  • 通讯作者:
    Lu Yang
WEAK SOLUTIONS AND INCOMPRESSIBLE LIMITS OF MULTI-DIMENSIONAL MAGNETOHYDRODYNAMIC FLOWS
多维磁流体流动的弱解与不可压缩极限
  • DOI:
  • 发表时间:
    2010
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Xianpeng Hu;M. Degree
  • 通讯作者:
    M. Degree
Global existence of weak solutions to two dimensional compressible viscoelastic flows

Xianpeng Hu的其他文献

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