Bourgain spaces
E547402
Bourgain spaces are function spaces introduced by Jean Bourgain that are tailored to study the well-posedness and regularity of nonlinear dispersive partial differential equations.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Banach space
ⓘ
function space ⓘ mathematical concept ⓘ |
| advantage |
allows sharp low-regularity results
ⓘ
captures dispersive smoothing effects ⓘ well-suited to Picard iteration ⓘ |
| alsoKnownAs |
Fourier restriction norm spaces
NERFINISHED
ⓘ
X^{s,b} spaces NERFINISHED ⓘ |
| appliedTo |
Korteweg–de Vries equations
NERFINISHED
ⓘ
general dispersive equations ⓘ non-periodic dispersive equations ⓘ nonlinear Schrödinger equations ⓘ periodic dispersive equations ⓘ |
| builtFrom |
dispersion relation ω(ξ)
ⓘ
solution operator of the linear dispersive equation ⓘ |
| characterizedBy |
adaptation to linear dispersive flows
ⓘ
dependence on spatial regularity index s ⓘ dependence on temporal regularity index b ⓘ frequency-time localization ⓘ space-time Fourier transform norms ⓘ use of modulation variable τ−ω(ξ) ⓘ weighted L2 norms in (ξ,τ)-space ⓘ |
| field |
dispersive PDE theory
ⓘ
functional analysis ⓘ harmonic analysis ⓘ partial differential equations ⓘ |
| introducedBy | Jean Bourgain NERFINISHED ⓘ |
| introducedInContextOf |
Korteweg–de Vries equation
NERFINISHED
ⓘ
nonlinear Schrödinger equation ⓘ periodic boundary conditions ⓘ |
| namedAfter | Jean Bourgain NERFINISHED ⓘ |
| normDefinedBy | L2 norm of weighted space-time Fourier transform ⓘ |
| property |
Banach space structure for fixed s and b
ⓘ
invariance under linear flow of the associated dispersive equation ⓘ |
| relatedTo |
Besov spaces
NERFINISHED
ⓘ
Sobolev spaces NERFINISHED ⓘ Strichartz spaces NERFINISHED ⓘ modulation spaces ⓘ |
| typicalNotation |
X^{s,b}
ⓘ
X^{s,b}_ ho for a dispersion relation ρ ⓘ |
| usedFor |
Fourier restriction norm method
NERFINISHED
ⓘ
Strichartz-type estimates NERFINISHED ⓘ bilinear and multilinear estimates ⓘ contraction mapping principle in PDE ⓘ fixed-point arguments in PDE ⓘ global well-posedness results ⓘ local well-posedness results ⓘ regularity theory for nonlinear PDEs ⓘ study of nonlinear dispersive partial differential equations ⓘ well-posedness of nonlinear PDEs ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.