# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Global regular solutions for Landau-Lifshitz equation. (English) Zbl 1222.35150
Summary: In this note, we prove that there exists a unique global regular solution for multidimensional Landau-Lifshitz equation if the gradient of solutions can be bounded in space ${L}^{2}\left(0,T;{L}^{\infty }\right)$. Moreover, for the two-dimensional radial symmetric Landau-Lifshitz equation with Neumann boundary condition in the exterior domain, this hypothesis in space ${L}^{2}\left(0,T;{L}^{\infty }\right)$ can be cancelled.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35K51 Second-order parabolic systems, initial bondary value problems
##### Keywords:
Landau-Lifshitz equation; global regular solution
##### References:
 [1] Landau L D, Lifshitz E M. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys Z Sowj, 1935, 8: 153–169. Reproduced in Collected Papers of Landau L D. New York: Pergamon Press, 1965. 101–114 [2] E Weinan, Wang Xiaoping Numerical methods for the Landau-Lifshitz equation. SIAM J Numer Anal, 2000, 38(5): 1647–1665 · Zbl 0988.65079 · doi:10.1137/S0036142999352199 [3] Fogedby H C. Theoretical Aspects of Mainly Low Dimensional Magnetic System. Lecture Notes Phys, 131. Berlin, Heidelberg, New York: Springer-Verlag, 1980 [4] Gustafson S, Shatah J. The stability of localize solutions of Landau-Lifshitz equations. Comm Pure Appl Math, 2002, 55(9): 1136–1159 · Zbl 1028.35059 · doi:10.1002/cpa.3024 [5] Hubert A, Schäfer R. Magnetic Domain–The Analysis of Magnetic Microstructures. Berlin, Heidelberg: Springer-Verlag, 1998 [6] Sulem P L, Sulem C, Bardos C. On the continuous limit for a system of classical spins. Comm Math Phys, 1986, 107: 431–454 · Zbl 0614.35087 · doi:10.1007/BF01220998 [7] Zhou Yulin, Guo Boling. The weak solution of homogeneous boundary value problem for the system of ferromegnetic chain with several variable. Scientia Sinica A, 1986, (4): 337–349 [8] Visintin A. On Landau-Lifshitz equations for ferromagnetism. Japan J Appl Math, 1985, 2: 69–84 · Zbl 0613.35018 · doi:10.1007/BF03167039 [9] Zhou Yulin, Guo Boling, Tan Shaobin. Existence and uniqueness of smooth solution for system of ferromagnetic chain. Science in China, Ser A, 1991, 34(3): 257–266 [10] Chang Nai-Heng, Shatah J, Uhlanbeck K. Schrödinger maps. Comm Pure Appl Math, 2000, 53(5): 590–602 · doi:10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R [11] Daniel M, Porsezian K, Lakshmanan M. On The integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions. J Math Phys, 1994, 35(12): 6498–6510 · Zbl 0821.35114 · doi:10.1063/1.530687 [12] Lakshmanan M, Porsezian K. Planar radially symmetric Heisenberg spin system, and generalized nonlinear Schrödinger equation: Gauge equivalence, Bäcklund transformations and explicit solutions. Phys Lett A, 1990, 146(6): 329–334 · doi:10.1016/0375-9601(90)90964-P [13] Guo Boling, Han Yongqian, Yang Ganshan. Blow up problem for Landau-Lifshitz equations in two dimensions. Comm Nonlinear Sci Numerical Simulation, 2000, 5(1): 43–44 · Zbl 0952.35143 · doi:10.1016/S1007-5704(00)90023-6 [14] Caffarelli L, Kohn R, Nirenberg L. First order interpolation inequalities with weights. Compositio Math, 1984, 53: 259–275 [15] Ladysenskaya O A. The Boundary Value Problems of Mathematical Physics. Applied Mathematical Sciences, 49. Berlin, Heidelberg, New York: Springer, 1985 [16] Zhou Yulin. Applications of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers, 1991