On the Cauchy problem for Schrödinger type equations and the regularity of the solutions

Doi, Satoshi

doi:10.1215/kjm/1250519013

Cited by 82 publications

(66 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Under some smoothness conditions on f, F and smoothness and decay assumptions on A, smooth solutions of (1.8) are known to exist globally; see for example Proposition 3.1 in Section 3, as well as earlier work of Doi, [7], [8]. These are all results based on L 2 a priori estimates for the corresponding equations.…”

Section: Introductionmentioning

confidence: 93%

See 1 more Smart Citation

On quadratic derivative Schrödinger equations in one space dimension

Stefanov

2007

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We consider the Schrödinger equation with derivative perturbation terms in one space dimension. For the linear equation, we show that the standard Strichartz estimates hold under specific smallness requirements on the potential. As an application, we establish existence of local solutions for quadratic derivative Schrödinger equations in one space dimension with small and rough Cauchy data.

show abstract

Section: Introductionmentioning

confidence: 93%

“…Most of these works were inspired by a pseudodifferential calculus approach pioneered by Doi, [7], [8]. This is due to the basic difficulty present in both (1.1) and (1.2), namely the lack of classical energy estimates.…”

Section: Introductionmentioning

confidence: 99%

On quadratic derivative Schrödinger equations in one space dimension

Stefanov

2007

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Thus we can easily diagonalize h(t,x,£;) and therefore this system becomes essentially same as two single equations. At the second step we apply S. Doi's method [3] to this diagonalized system.…”

Section: ) Where V (T X} =' (Vi (T X} V 2 (T X)) Is C 2 -Valuedmentioning

confidence: 99%

“…Thus we can apply the theory of linear Schrodinger type equations. The following arguments in this subsection are basically due to S. Doi [3].…”

Section: (/I(t)v} +I(a +B Diag «) U(t) V ) + (C(t) -A T (T)}v=a(t)fmentioning

confidence: 99%

“…It seems to be defficult to extend their results to global existence. On the other hand, S. Doi [3] established a rather simple and useful method for linear Schrodinger type equations. Actually, in [2] , using his method with some modifications, we proved the local existence theorem for general semilinear equations with large initial data.…”

Section: §1 Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Global Existence of Small Solutions to Semilinear Schrödinger Equations with Gauge Invariance

Chihara

1995

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

show abstract

Erratum: On Schrödinger maps

Nahmod

Stefanov

Uhlenbeck

2004

Comm Pure Appl Math

View full text Add to dashboard Cite

In our paper [11] there is a serious error in the translation from the geometric equation for Schrödinger maps to the analytic equation after the gauge has been fixed. The Schrödinger map equation is written in the Coulomb gauge aswhere a is real and F are lower-order terms. In the translation, this was misinterpreted and the main estimate applies only towhere F are the same lower-order terms as above. Hence, the results of Theorem 4.1 in [11], while correct for system (2), do not apply to the Schrödinger map. Except for the work of Sulem, Sulem, and Bardos [13], the only results for Schrödinger maps with data in the classical Sobolev spaces up until now come essentially from energy methods. These are described under various regularity assumptions on the data by Doi [4,5] Kenig, Ponce, and Vega [8,9].The best possible result along this line presently available gives existence and uniqueness for data in H 2+ε , ε > 0, not in H 1+ε as claimed in the paper [11]. We give a brief outline of a proof for the sphere for simplicity (this can certainly be extended to more general targets).Energy methods consist of assuming smooth solutions, giving a priori energy estimates taking weak limits, and proving uniqueness as it was done in the paper [11]. We give here the two key a priori estimates.

show abstract

On the Cauchy problem for Schrödinger type equations and the regularity of the solutions

Cited by 82 publications

References 0 publications

On quadratic derivative Schrödinger equations in one space dimension

On quadratic derivative Schrödinger equations in one space dimension

Global Existence of Small Solutions to Semilinear Schrödinger Equations with Gauge Invariance

Erratum: On Schrödinger maps

Contact Info

Product

Resources

About