量子Navier-Stokes-Landau-Lifshitz方程光滑解的爆破问题

数学物理学报 ›› 2022, Vol. 42 ›› Issue (4): 1074-1088.

量子Navier-Stokes-Landau-Lifshitz方程光滑解的爆破问题

邱臻*(),王光武

广州大学数学与信息科学学院广州 510006

收稿日期:2021-06-12 出版日期:2022-08-26 发布日期:2022-08-08
通讯作者: 邱臻 E-mail:qiuzhen1997826@qq.com
基金资助:
国家自然科学基金(11801107);广州市科技计划项目-市校(院)联合资助项目(202102010467)

Blow-Up of the Smooth Solutions to the Quantum Navier-Stokes-Landau-Lifshitz Equations

Zhen Qiu*(),Guangwu Wang

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006

Received:2021-06-12 Online:2022-08-26 Published:2022-08-08
Contact: Zhen Qiu E-mail:qiuzhen1997826@qq.com
Supported by:
the NSFC(11801107);the Science and Technology Program of Guangzhou, China(202102010467)

摘要/Abstract

摘要：

该文研究量子Navier-Stokes-Landau-Lifshitz(QNSLL)方程组在区域$\Omega \subseteq \mathbb{R} ^n$ $(n=1, 2)$上光滑解的爆破问题, 证明了QNSLL方程组在上半空间$\mathbb{R} _+^n$、全空间$\mathbb{R}^n$以及球形区域上的光滑解将在有限时间内爆破, 其中上半空间$\mathbb{R} _+^n$、全空间$\mathbb{R}^n$上的局部光滑解的爆破时间依赖于边界条件, 球形区域的局部光滑解的爆破时间则依赖于边界条件和初值条件.特别地, 以上结论对NSLL方程组也成立.

关键词: 量子Navier-Stokes-Landau-Lifshitz方程, 局部光滑解, 爆破时间

Abstract:

In this paper, we investigate the blow-up of the smooth solutions to the quantum Navier-Stokes-Landau-Lifshitz systems(QNSLL) in the domains $\Omega \subseteq \mathbb{R} ^n(n =1, 2)$. We prove that the smooth solutions to the QNSLL will blow up in finite time in the domains half-space $\mathbb{R} _+^n$, whole-space $\mathbb{R} ^n$ and ball. The paper also shows that the blow-up time of the smooth solutions in half-space or whole-space only depends on boundary conditions, while the the blow-up time of the smooth solutions in the ball depends on initial data and boundary conditions. In particular, the above conclusions are also valid for NSLL systems.

Key words: Quantum Navier-Stokes-Landau-Lifshitz equations, Smooth Solution, Blow-up

中图分类号:

O175.29

邱臻,王光武. 量子Navier-Stokes-Landau-Lifshitz方程光滑解的爆破问题[J]. 数学物理学报, 2022, 42(4): 1074-1088.

Zhen Qiu,Guangwu Wang. Blow-Up of the Smooth Solutions to the Quantum Navier-Stokes-Landau-Lifshitz Equations[J]. Acta mathematica scientia,Series A, 2022, 42(4): 1074-1088.

参考文献

1	Alouges F , Soyeur A . On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Anal, 1992, 18 (11): 1071- 1084
2	Antonelli P , Spirito S . Global existence of finite energy weak solutions of quantum Navier-Stokes equations. Arch Ration Mech Anal, 2017, 225 (3): 1161- 1199
3	Antonelli P , Spirito S . On the compactness of finite energy weak solutions to the quantum Navier-Stokes equations. Journal of Hyperbolic Differential Equations, 2018, 15 (1): 133- 147
4	Bene?ová B , Forster J , Liu C , et al. Schl?merkemper A. Existence of weak solutions to an evolutionary model for magnetoelasticity. SIAM J Math Anal, 2018, 50 (1): 1200- 1236
5	Chen Y M , Ding S J , Guo B L . Partial regularity for two-dimensional Landau-Lifshitz equations. Acta Math Sinica (NS), 1998, 14 (3): 423- 432
6	Ding S J , Wang C Y . Finite time singularity of the Landau-Lifshitz-Gilbert equation. Int Math Res Not IMRN, 2007, 25 (4): Art ID rnm012
7	Feireisl E . Dynamics of Viscous Compressible Fluids. Volume 26 of Oxford Lecture Series in Mathematics and its Applications. Oxford: Oxford University Press, 2004
8	Feireisl E , Novotny A , Petzeltová H . On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3 (4): 358- 392
9	Gamba I M , Gualdani M P , Zhang P . On the blowing up of solutions to quantum hydrodynamic models on bounded domains. Monatsh Math, 2009, 157 (1): 37- 54
10	Gamba I M , Jüngel A . Positive solutions to singular second and third order differential equations for quantum fluids. Arch Ration Mech Anal, 2001, 156 (3): 183- 203
11	Gamba I M , Jüngel A . Asymptotic limits for quantum trajectory models. Comm Partial Differential Equations, 2002, 27 (3/4): 669- 691
12	Gisclon M , Lacroix-Violet I . About the barotropic compressible quantum Navier-Stokes equations. Nonlinear Anal, 2015, 128: 106- 121
13	Gualdini M P , Jüngel A . Analysis of the viscous quantum hydrodynamic equations for semiconductors. European J Appl Math, 2004, 15 (5): 577- 595
14	Guo B L , Hong M C . The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps. Calc Var Partial Differential Equations, 1993, 1 (3): 311- 334
15	Guo B L, Ding S J. Landau-Lifshitz Equations, Volume 1 of Frontiers of Research with the Chinese Academy of Sciences. Hackensack, NJ: World Scientific Publishing Co Pte Ltd, 2008
16	Jüngel A . Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J Math Anal, 2010, 42 (3): 1025- 1045
17	Jüngel A , Mili?i? J P . Physical and numerical viscosity for quantum hydrodynamics. Commun Math Sci, 2007, 5 (2): 447- 471
18	Kalousek M , Kortum J , Schl?merkemper A . Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete Contin Dyn Syst Ser S, 2021, 14 (1): 17- 39
19	Lions P L . Mathematical Topics in Fluid Mechanics. New York: Oxford University Press, 1998
20	Vasseur A F , Cheng Yu . Global weak solutions to the compressible quantum Navier-Stokes equations with damping. SIAM J Math Anal, 2016, 48 (2): 1489- 1511
21	Wang G W , Guo B L . A Blow-up criterion of strong solutions to the quantum hydrodynamic Model. Acta Math Sci, 2020, 40B (3): 795- 804
22	Wang G W , Guo B L . Blow-up of the smooth solution to quantum hydrodynamic models in $\mathbf{R}^d$. J Differential Equations, 2016, 261 (7): 3815- 3842
23	Wang G W , Guo B L . Blow-up of solutions to quantum hydrodynamic models in half space. J Math Phys, 2017, 58 (3): 031505
24	Wang C Y . On Landau-Lifshitz equation in dimensions at most four. Indiana Univ Math J, 2006, 55 (5): 1615- 1644
25	Wang G W , Guo B L . Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete Contin Dyn Syst Ser B, 2019, 24 (11): 6141- 6166
26	Wang G W , Guo B L . A new blow-up criterion of the strong solution to the quantum hydrodynamic model. Appl Math Lett, 2021, 119: 107045, 7
27	Wang G W , Guo B L , Fang S M . Blow-up of the smooth solutions to the compressible Navier-Stokes equations. Math Methods Appl Sci, 2017, 40 (14): 5262- 5272
28	Xin Z P . Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm Pure Appl Math, 1998, 51 (3): 229- 240
29	Zhou Y L , Guo B L , Tan S B . Existence and uniqueness of smooth solution for system of ferromagnetic chain. Science in China, Ser A, 1991, 34 (3): 257- 266

[1]	李锋杰, 李平. 伪抛物型 p-Kirchhoff 方程的爆破解[J]. 数学物理学报, 2024, 44(3): 717-736.
[2]	杨慧,韩玉柱. 一类抛物型Kirchhoff方程解的爆破性质[J]. 数学物理学报, 2021, 41(5): 1333-1346.
[3]	郑亚东,方钟波. 一类具有时变系数梯度源项的弱耦合反应-扩散方程组解的爆破分析[J]. 数学物理学报, 2020, 40(3): 735-755.
[4]	马丹旎,方钟波. 具有耦合指数反应项的变系数扩散方程组解的爆破现象[J]. 数学物理学报, 2019, 39(6): 1456-1475.
[5]	龙群飞,陈建清. 一类带梯度依赖势和源的粘性Cahn-Hilliard方程的爆破现象[J]. 数学物理学报, 2019, 39(3): 510-517.
[6]	孙宝燕. 具有非局部源的p-Laplace方程解的爆破时间下界估计[J]. 数学物理学报, 2018, 38(5): 911-923.
[7]	赵元章, 马相如. 一类具有变扩散系数的非局部反应-扩散方程解的爆破分析[J]. 数学物理学报, 2018, 38(4): 750-769.
[8]	马羚未, 方钟波. 具有加权非局部源和Robin边界条件的反应-扩散方程解的爆破时间下界[J]. 数学物理学报, 2017, 37(1): 146-157.
[9]	凌征球, 王泽佳. 带具有耗散梯度项的p-Laplace方程解的爆破问题[J]. 数学物理学报, 2016, 36(5): 958-964.