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学者姓名：林雪云
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In this paper, we study the well-posedness theory of the magneto-micropolar boundary layer and justify the high Reynolds numbers limit for the magneto-micropolar system with Prandtl boundary layer expansion. If the initial tangential magnetic field is nondegenerate, we obtain the local-in-time existence, uniqueness of solutions for the incompressible magneto-micropolar boundary layer equations with the lower regularity initial data in H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_3$$\end{document}. This work is inspired by the recent progresses by Liu et al. (Commun Pure Appl Math 72(1):63-121, 2019) for 2D MHD boundary layer equations theory in Sobolev spaces without monotonicity. There are some differences between this work and Liu et al. (2019). First, the model has the familar form of the MHD equations but it coupled with the equation of the microrotation field, which essentially describes the motion inside the macrovolume as they undergo mocro-rotational effects represented by the micro-rotational velocity vector. Second, the lack of higher-order boundary condition at y=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document} is one of the main difficulties to solve the Prandtl type equations. We present a reconstruction argument for higher-order boundary conditions to fix this technical difficulty. Third, our greatest difference is the application of a stratified energy estimates instead of using space-time differential multi-indices as in Liu et al. (2019), so we can work in a functional framework of lower regularity. Our analysis is based on direct Sobolev spaces and allows us to give an L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>\infty $$\end{document} estimate on the error by multiscale analysis under the assumption that the kinematic, spin viscosity coefficients and the resistivity coefficient are of the same order and the initial tangential magnetic field on the boundary is not degenerate.
Keyword ：
35B30 35B30 35M33 35M33 35Q35 35Q35 76N20 76N20 76W05 76W05
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| GB/T 7714 | Lin, Xue-yun , Liu, Cheng-jie , Zhang, Ting . Magneto-micropolar boundary layers theory in Sobolev spaces without monotonicity: well-posedness and convergence theory [J]. | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS , 2024 , 63 (3) . |
| MLA | Lin, Xue-yun 等. "Magneto-micropolar boundary layers theory in Sobolev spaces without monotonicity: well-posedness and convergence theory" . | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 63 . 3 (2024) . |
| APA | Lin, Xue-yun , Liu, Cheng-jie , Zhang, Ting . Magneto-micropolar boundary layers theory in Sobolev spaces without monotonicity: well-posedness and convergence theory . | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS , 2024 , 63 (3) . |
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In this paper, we establish the well-posedness theory for the three-dimensional axially symmetric magnetohydrodynamic (MHD) boundary layer system in Gevrey function space without any structural assumption. By using a refined cancellation mechanism to overcome the loss of tangential derivatives in the system and constructing a refined energy functional involves in a polynomial weight on the tangential variables to overcome the order mismatch between the tangentially radial field and the normal field, we show that the three-dimensional axially symmetric MHD boundary layer system is well-posed with Gevrey index up to 3/2. Our result is an extension of the previous work (Li and Yang in SIAM J Math Anal 53(3):3236-3264, 2021) from the MHD boundary layer system in both two- and three-dimensional spaces to the axisymmetric case.
Keyword ：
3D axially symmetric MHD boundary layer 3D axially symmetric MHD boundary layer Gevrey class Gevrey class nonstructural assumption nonstructural assumption well-posedness theory well-posedness theory
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| GB/T 7714 | Lin, Xueyun , Zou, Lin . Well-Posedness in Gevrey Function Space for the 3D Axially Symmetric MHD Boundary Layer Equations Without Structural Assumption [J]. | RESULTS IN MATHEMATICS , 2024 , 79 (2) . |
| MLA | Lin, Xueyun 等. "Well-Posedness in Gevrey Function Space for the 3D Axially Symmetric MHD Boundary Layer Equations Without Structural Assumption" . | RESULTS IN MATHEMATICS 79 . 2 (2024) . |
| APA | Lin, Xueyun , Zou, Lin . Well-Posedness in Gevrey Function Space for the 3D Axially Symmetric MHD Boundary Layer Equations Without Structural Assumption . | RESULTS IN MATHEMATICS , 2024 , 79 (2) . |
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This paper focuses on the Rayleigh-Taylor instability in the system of equations of the two-dimensional nonhomogeneous incompressible Euler-Korteweg equations in a horizontal periodic domain with infinite height. First, we use variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh- Taylor problem. Then, motivated by the Grenier's idea in [21], we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh- Taylor problem due to the absence of viscosity in the system, and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh-Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang-Guo in [28], and thus obtain the nonlinear Rayleigh-Taylor instability result, which presents that the Rayleigh- Taylor instability can occur in the capillary fluids for any capillary coefficient kappa > 0 if the critical capillary number is infinite. (c) 2022 Elsevier Inc. All rights reserved.
Keyword ：
Incompressible Incompressible Incompressible capillary fluids Incompressible capillary fluids Navier-Stokes-Korteweg equations Navier-Stokes-Korteweg equations Rayleigh-Taylor instability Rayleigh-Taylor instability
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| GB/T 7714 | Zhang, Xuyan , Hua, Zhiwei , Jiang, Han et al. On the dynamic Rayleigh-Taylor instability in the Euler-Korteweg model [J]. | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2023 , 520 (2) . |
| MLA | Zhang, Xuyan et al. "On the dynamic Rayleigh-Taylor instability in the Euler-Korteweg model" . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 520 . 2 (2023) . |
| APA | Zhang, Xuyan , Hua, Zhiwei , Jiang, Han , Lin, Xueyun . On the dynamic Rayleigh-Taylor instability in the Euler-Korteweg model . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2023 , 520 (2) . |
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In this paper, we investigate the well-posedness and large time behavior of solutions to the 3D incompressible magneto-micropolar equations. By virtue of the L-p - L-q estimate obtained through the spectral decomposition of the linearized magneto-micropolar equations, we show the existence and uniqueness of small L-3-strong solutions of the equations with small initial data. Then basing on this result, we derive sharp time decay estimates of the L-3-strong solutions.
Keyword ：
3D magneto-micropolar equations 3D magneto-micropolar equations banach contraction mapping principle banach contraction mapping principle large time decay large time decay spectral decomposition spectral decomposition
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| GB/T 7714 | Ye, Xiuping , Lin, Xueyun . Decay estimates of the 3D magneto-micropolar system with applications to L3-strong solutions [J]. | APPLICABLE ANALYSIS , 2023 , 103 (11) : 1903-1921 . |
| MLA | Ye, Xiuping et al. "Decay estimates of the 3D magneto-micropolar system with applications to L3-strong solutions" . | APPLICABLE ANALYSIS 103 . 11 (2023) : 1903-1921 . |
| APA | Ye, Xiuping , Lin, Xueyun . Decay estimates of the 3D magneto-micropolar system with applications to L3-strong solutions . | APPLICABLE ANALYSIS , 2023 , 103 (11) , 1903-1921 . |
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In this paper, we consider the vanishing viscosity limit for the incompressible non-resistive magneto-micropolar equations on the half-space with no-slip boundary condition (3). We prove that the vanishing viscosity limit is uniform over a time interval, which indicates that the incompressible non-resistive magneto-micropolar equations with the no-slip boundary condition have a strong solution and the solution is uniformly bounded in both the conormal Sobolev norm and L-infinity norm. As a direct result, we obtain the vanishing viscosity limit for the incompressible non-resistive magneto-micropolar equations by a strong compactness argument.
Keyword ：
Incompressible non-resistive magneto-micropolar equations Incompressible non-resistive magneto-micropolar equations no-slip boundary condition no-slip boundary condition uniform regularity uniform regularity vanishing viscosity limit vanishing viscosity limit
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| GB/T 7714 | Zou, Lin , Lin, Xueyun . Uniform regularity and vanishing viscosity limit for the incompressible non-resitive magneto-micropolar equations [J]. | APPLICABLE ANALYSIS , 2022 . |
| MLA | Zou, Lin et al. "Uniform regularity and vanishing viscosity limit for the incompressible non-resitive magneto-micropolar equations" . | APPLICABLE ANALYSIS (2022) . |
| APA | Zou, Lin , Lin, Xueyun . Uniform regularity and vanishing viscosity limit for the incompressible non-resitive magneto-micropolar equations . | APPLICABLE ANALYSIS , 2022 . |
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In this paper, we consider the magnetic effect on the Sobolev solvability of the two-dimensional incompressible magneto-micropolar boundary layer system without resistivity. This gives a complement to the previous work of Lin et al. (2022), where the well-posedness and the convergence theory were established for the magneto-micropolar boundary layer system without monotonicity in Sobolev spaces. If the initial tangential magnetic field is not degenerate, a local-in-time well-posedness theory in Sobolev spaces is established without the monotonic condition on the velocity or the micro-rotational velocity. Moreover, when the tangential magnetic field of shear layer degenerates at the non-degenerate critical point of the initial velocity and the initial micro-rotational velocity, the linearized magneto-micropolar boundary layer system around a shear flow with general decay is ill-posed in Sobolev spaces. (C) 2022 Elsevier Ltd. All rights reserved.
Keyword ：
General decay General decay Ill-posedness Ill-posedness Magneto-micropolar boundary layer Magneto-micropolar boundary layer Sobolev spaces Sobolev spaces Well-posedness Well-posedness
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| GB/T 7714 | Zou, Lin , Lin, Xueyun . Magnetic effects on the solvability of 2D incompressible magneto-micropolar boundary layer equations without resistivity in Sobolev spaces [J]. | NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS , 2022 , 224 . |
| MLA | Zou, Lin et al. "Magnetic effects on the solvability of 2D incompressible magneto-micropolar boundary layer equations without resistivity in Sobolev spaces" . | NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 224 (2022) . |
| APA | Zou, Lin , Lin, Xueyun . Magnetic effects on the solvability of 2D incompressible magneto-micropolar boundary layer equations without resistivity in Sobolev spaces . | NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS , 2022 , 224 . |
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We investigate the nonlinear Rayleigh-Taylor (RT) instability of a wnonhomogeneous incompressible nematic liquid crystal in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution. Thus we construct solutions of the linearized problem that grow in time in the Sobolev space H-4, then we show that the RT equilibrium state is linearly unstable. With the help of the established unstable solutions of the linearized problem and error estimates between the linear and nonlinear solutions, we establish the nonlinear instability of the density, the horizontal and vertical velocities under L-1-norm.
Keyword ：
Liquid crystals Liquid crystals Rayleigh-Taylor instability Rayleigh-Taylor instability Steady state solution Steady state solution
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| GB/T 7714 | Liu, Mengmeng , Lin, Xueyun . On instability of Rayleigh-Taylor problem for incompressible liquid crystals under L-1-norm [J]. | BOUNDARY VALUE PROBLEMS , 2021 , 2021 (1) . |
| MLA | Liu, Mengmeng et al. "On instability of Rayleigh-Taylor problem for incompressible liquid crystals under L-1-norm" . | BOUNDARY VALUE PROBLEMS 2021 . 1 (2021) . |
| APA | Liu, Mengmeng , Lin, Xueyun . On instability of Rayleigh-Taylor problem for incompressible liquid crystals under L-1-norm . | BOUNDARY VALUE PROBLEMS , 2021 , 2021 (1) . |
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In this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.
Keyword ：
Damping mechanism Damping mechanism Elastodynamics Elastodynamics Global classical solution Global classical solution Time-weighted energy estimate Time-weighted energy estimate
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| GB/T 7714 | Liu, Mengmeng , Lin, Xueyun . Global classical solutions to the elastodynamic equations with damping [J]. | JOURNAL OF INEQUALITIES AND APPLICATIONS , 2021 , 2021 (1) . |
| MLA | Liu, Mengmeng et al. "Global classical solutions to the elastodynamic equations with damping" . | JOURNAL OF INEQUALITIES AND APPLICATIONS 2021 . 1 (2021) . |
| APA | Liu, Mengmeng , Lin, Xueyun . Global classical solutions to the elastodynamic equations with damping . | JOURNAL OF INEQUALITIES AND APPLICATIONS , 2021 , 2021 (1) . |
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In this paper, we consider the large time behavior of the Cauchy problem for the three-dimensional isentropic compressible magnetohydrodynamic (MHD) equations. The global existence of smooth solutions for the 3D compressible MHD equations has been proved by Chen-Tan [6], under the condition that the initial data are close to the constant equilibrium state in the Sobolev space H-3. However, to our best knowledge, the decay estimate of the highest-order spatial derivatives of the solution to the compressible MHD equations has not been solved. The main goal in this paper is to give a positive answer to this problem. Exactly, under the assumption that the initial perturbation is small in H-l(R-3) boolean AND (B) over dot(2,infinity)(-s)(R-3) with l >= 3, s is an element of inverted right perpendicular0,5/2inverted left perpendicular, combining the spectral analysis on the semigroup generated by the linear system at the constant state and the energy method to the compressible MHD equations, then we get the optimal convergence rates of any order spatial derivatives (including the highest-order derivatives) of the solution. This result concern with the optimal time decay rates of the solutions, which extends the work obtained by Chen [5] for the compressible Navier-Stokes equations in R-3 to the 3D compressible MHD equations. Moreover, we have expanded the range of sfrom (0, 3/2] to [0, 5/2], compared with the previous results on optimal decay rates of global strong solutions for the 3D isentropic compressible MHD system. (C) 2021 Elsevier Inc. All rights reserved.
Keyword ：
Compressible MHD equations Compressible MHD equations Highest-order derivatives Highest-order derivatives Optimal time decay rates Optimal time decay rates Semigroup theory Semigroup theory
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| GB/T 7714 | Huang, Wenting , Lin, Xueyun , Wang, Weiwei . Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations [J]. | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2021 , 502 (2) . |
| MLA | Huang, Wenting et al. "Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations" . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 502 . 2 (2021) . |
| APA | Huang, Wenting , Lin, Xueyun , Wang, Weiwei . Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations . | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS , 2021 , 502 (2) . |
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In this paper, we prove the almost global existence of classical solutions to the 3D Prandtl system with the initial data which lie within epsilon of a stable shear flow. Using anisotropic Littlewood-Paley energy estimates in tangentially analytic norms and introducing new linearly-good unknowns, we prove that the 3D Prandtl system has a unique solution with the lifespan of which is greater than exp(epsilon(-1)/ log(epsilon(-1))). This result extends the work obtained by Ignatova and Vicol (Arch. Ration. Mech. Anal. 2:809-848, 2016) on the 2D Prandtl equations to the three-dimensional setting.
Keyword ：
Almost global existence Almost global existence Littlewood-Paley theory Littlewood-Paley theory Prandtl equations Prandtl equations
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| GB/T 7714 | Lin, Xueyun , Zhang, Ting . Almost Global Existence for the 3D Prandtl Boundary Layer Equations [J]. | ACTA APPLICANDAE MATHEMATICAE , 2019 , 169 (1) : 383-410 . |
| MLA | Lin, Xueyun et al. "Almost Global Existence for the 3D Prandtl Boundary Layer Equations" . | ACTA APPLICANDAE MATHEMATICAE 169 . 1 (2019) : 383-410 . |
| APA | Lin, Xueyun , Zhang, Ting . Almost Global Existence for the 3D Prandtl Boundary Layer Equations . | ACTA APPLICANDAE MATHEMATICAE , 2019 , 169 (1) , 383-410 . |
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