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 Schrödinger maps in dimensions d2: small data in the critical Sobolev spaces. (English) Zbl 1233.35112

Two global results for the initial-value problem for the Schrödinger map equation, t φ=φ×Δφ, on d ×, φ(0)=φ 0 , φ: d ×𝕊 2 , d2, are proved. In order to formulate these results we have to introduce some notation. As usual, H σ denotes the Sobolev space on d . For Q𝕊 2 , H Q σ ={f: d 3) ;|f(x)|=1,a.e.,f-QH σ }, f H Q σ =f-Q H σ . The space H Q is the intersection of all spaces H Q σ and for fH , f H ˙ σ is the homogeneous Sobolev norm.

The first theorem proved in the paper states that if d2 and Q𝕊 2 , then there exist ε 0 (d)>0 and a constant C>0 such that for any φ 0 H Q σ with φ 0 -Q H ˙ d/2 ε 0 (d) there exists a unique solution φ=S Q (φ 0 )C(;H Q ) of the initial-value problem, φ(t)-Q H ˙ d/2 Cφ 0 -Q H ˙ d/2 ,forallt and for any T[0,) and σ + , sup t[-T,T] φ(t) H Q σ C(σ,T,φ 0 H Q σ ).

The second theorem provides uniform bounds (on ) for φ(t) H Q σ and asserts that the operator S Q admits continuous extensions S Q :B ε 0 (d,σ 1 ) σ C(;H ˙ σ H ˙ Q d/2-1 ) for any σ[d/2,σ 1 ] and some ε 0 (d,σ 1 )(0,ε 0 (d)), where B ε σ ={φH ˙ Q d/2-1 H ˙ σ ;φ-Q| H ˙ d/2 ε}.

The first theorem was proved in dimensions d4 in [I. Bejenaru, A. D. Ionescu, and C. E. Kenig, Adv. Math. 215, No. 1, 263–291 (2007; Zbl 1152.35049)]. In order to overcome the difficulties which appear, especially in the case d=2, the authors of the paper under review use a sum of Galilei transforms of the lateral L e p,q spaces and the caloric gauge.

MSC:
35K45Systems of second-order parabolic equations, initial value problems
35B65Smoothness and regularity of solutions of PDE
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35Q41Time-dependent Schrödinger equations, Dirac equations