A para-differential renormalization technique for nonlinear dispersive equations

Sebastian Herr; Alexandru D. Ionescu; Carlos E. Kenig; Herbert Koch

doi:10.1080/03605302.2010.487232

A para-differential renormalization technique for nonlinear dispersive equations

Sebastian Herr, Alexandru D. Ionescu, Carlos E. Kenig, Herbert Koch

Research output: Contribution to journal › Article › peer-review

48 Scopus citations

Abstract

For α ∈ (1, 2) we prove that the initial-value problem {∂_tu + D^α∂_xu + ∂_x(u²/2) = 0 on R_x × R_t; u(0) =, is globally well-posed in the space of real-valued L²-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.

Original language	English (US)
Pages (from-to)	1827-1875
Number of pages	49
Journal	Communications in Partial Differential Equations
Volume	35
Issue number	10
DOIs	https://doi.org/10.1080/03605302.2010.487232
State	Published - 2010
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

Keywords

Dispersion generalized benjamin-ono equation
Frequencydependent renormalization
Global well-posedness
L initial data

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10.1080/03605302.2010.487232

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title = "A para-differential renormalization technique for nonlinear dispersive equations",

abstract = "For α ∈ (1, 2) we prove that the initial-value problem {∂tu + Dα∂xu + ∂x(u2/2) = 0 on Rx × Rt; u(0) =, is globally well-posed in the space of real-valued L2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.",

keywords = "Dispersion generalized benjamin-ono equation, Frequencydependent renormalization, Global well-posedness, L initial data",

author = "Sebastian Herr and Ionescu, {Alexandru D.} and Kenig, {Carlos E.} and Herbert Koch",

note = "Funding Information: S. H. was supported in part by the DFG, Sonderforschungsbereich 611. A. I. was supported in part by a Packard Fellowship. C. K. was supported in part by NSF grant DMS0456583. H. K. was supported in part by the DFG, Sonderforschungsbereich 611.",

year = "2010",

doi = "10.1080/03605302.2010.487232",

language = "English (US)",

volume = "35",

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journal = "Communications in Partial Differential Equations",

issn = "0360-5302",

publisher = "Taylor and Francis Ltd.",

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TY - JOUR

T1 - A para-differential renormalization technique for nonlinear dispersive equations

AU - Herr, Sebastian

AU - Ionescu, Alexandru D.

AU - Kenig, Carlos E.

AU - Koch, Herbert

N1 - Funding Information: S. H. was supported in part by the DFG, Sonderforschungsbereich 611. A. I. was supported in part by a Packard Fellowship. C. K. was supported in part by NSF grant DMS0456583. H. K. was supported in part by the DFG, Sonderforschungsbereich 611.

PY - 2010

Y1 - 2010

N2 - For α ∈ (1, 2) we prove that the initial-value problem {∂tu + Dα∂xu + ∂x(u2/2) = 0 on Rx × Rt; u(0) =, is globally well-posed in the space of real-valued L2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.

AB - For α ∈ (1, 2) we prove that the initial-value problem {∂tu + Dα∂xu + ∂x(u2/2) = 0 on Rx × Rt; u(0) =, is globally well-posed in the space of real-valued L2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.

KW - Dispersion generalized benjamin-ono equation

KW - Frequencydependent renormalization

KW - Global well-posedness

KW - L initial data

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DO - 10.1080/03605302.2010.487232

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SP - 1827

EP - 1875

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 10

ER -

A para-differential renormalization technique for nonlinear dispersive equations

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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