数学研究

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研究兴趣

    我主要研究流体力学中的偏微分方程，尤其是无粘流体的自由边界演化问题，包括水波、磁流体、
    弹性力学、相对论流体等。目前的研究关注如下方面：
  • 可压缩流体的（带/不带表面张力）两相流自由界面问题的适定性。例如涡片解（vortex sheet）、接触间断（contact discontinuity）。这可以视作一阶对称双曲组的强特征间断的稳定性。
  • 自由边界可压缩无粘流的不可压极限，包括初值不是Well-prepared的情况。这可以视为具有特征（自由）边界的一阶对称双曲组的奇异极限问题。
  • 自由边界无粘流体方程的奇异性或长时间解。例如有限时间爆破、小初值长时间解、或是某些稳态解的稳定性等等。
发表文献与预印本

我的所有文章可以在arXiv或者researchgate里面找到预印本。

  • [12] Junyan Zhang. Nonlinear Stability and Incompressible Limit of 3D Current-Vortex Sheets with or without Surface Tension in Ideal MHD. In preparation.
  • [11] Jiawei Wang, Junyan Zhang. Incompressible Limit of Compressible Ideal MHD with Tangential Magnetic Fields. In preparation.
  • [10] Chenyun Luo, Junyan Zhang. Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-posedness, Incompressible and Zero-Surface-Tension Limits. arXiv: 2211.03600, preprint. [PDF].

  • 博士毕业论文: Junyan Zhang. The Free-Boundary Problems in Inviscid Magnetohydrodynamics with or without Surface Tension. (292页，实为我写的MHD文章[1-3,5,7-9]的总结性论文,其中修正了[5]中的一些小错误.) [DOI] [PDF].

  • [9] Xumin Gu, Chenyun Luo, Junyan Zhang. Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics. Nonlinearity, 35(12), 6349-6398 (2022). arXiv: 2109.05400. [DOI] [PDF].
  • [8] Hans Lindblad, Junyan Zhang. Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics. arXiv: 2106.12173, preprint. [PDF].
  • [7] Xumin Gu, Chenyun Luo, Junyan Zhang. Local well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension. Accepted by J. Math Pures Appl. arXiv: 2105.00596. [PDF].
  • [6] Junyan Zhang. Local well-posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics. Arch. Rational Mech. Anal., 244(3), 599-697 (2022). arXiv: 2102.07979. [DOI] [PDF].
  • [5] Junyan Zhang. Local well-posedness of the Free-Boundary Problem in Compressible Resistive Magnetohydrodynamics. Calc. Var. Partial Differ. Equ., 62(4): 124. (2023) arXiv: 2012.13931. [DOI] [PDF].
  • [4] Chenyun Luo, Junyan Zhang. Local Well-posedness for the Motion of a Compressible Gravity Water Wave with Vorticity. J. Differ. Eq., Vol.332, 333-403 (2022). arXiv: 2109.02822. (本文初稿于2020年4月12日完成并首次投稿) [DOI] [PDF]
  • [3] Junyan Zhang. A priori Estimates for the free-boundary problem of Compressible Resistive MHD Equations and Incompressible Limit. arXiv: 1911.04928, preprint. [PDF]
  • [2] Chenyun Luo, Junyan Zhang. A priori Estimates for the Incompressible Free-Boundary Magnetohydrodynamics Equations with Surface Tension. SIAM J. Math. Anal., 53(2), 2595-2630 (2021). arXiv: 1907.11827. [DOI] [PDF]
  • [1] Chenyun Luo, Junyan Zhang. A Regularity Result for the Incompressible Magnetohydrodynamics Equations with Free Surface Boundary. Nonlinearity, 33(4), 1499–1527 (2020). arXiv: 1904.05444 . [DOI] [PDF]



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