# Difference between revisions of "Wave estimates"

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<center><math>\| u \|_{X^{s,b}} = \| <\xi>^s <|\xi| - |\tau|>^b \hat{u} ( \tau, \xi )\|_2 </math></center> | <center><math>\| u \|_{X^{s,b}} = \| <\xi>^s <|\xi| - |\tau|>^b \hat{u} ( \tau, \xi )\|_2 </math></center> | ||

− | Linear space-time estimates are known as [[Strichartz estimates]]. They are especially useful for the [[NLW|semilinear NLW without derivatives]], and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The <math>X^{s,b}_{}</math> spaces are used primarily for [[bilinear wave estimates|bilinear estimates], although more recently [[multilinear wave estimates|multilinear estimates have begun to appear]]. These spaces first appear in one-dimension in [[RaRe1982]] and in higher dimensions in [[Be1983]] in the context of [[propagation of singularities]]; they were used implicitly for LWP in [[KlMa1993]], while the Schrodinger and KdV analogues were developed in [[Bo1993]], [[Bo1993b]]. | + | Linear space-time estimates are known as [[Strichartz estimates]]. They are especially useful for the [[NLW|semilinear NLW without derivatives]], and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The <math>X^{s,b}_{}</math> spaces are used primarily for [[bilinear wave estimates|bilinear estimates]], although more recently [[multilinear wave estimates|multilinear estimates have begun to appear]]. These spaces first appear in one-dimension in [[RaRe1982]] and in higher dimensions in [[Be1983]] in the context of [[propagation of singularities]]; they were used implicitly for LWP in [[KlMa1993]], while the Schrodinger and KdV analogues were developed in [[Bo1993]], [[Bo1993b]]. |

== Specific wave estimates == | == Specific wave estimates == |

## Revision as of 16:37, 11 December 2009

Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms , or in spaces, defined by

Linear space-time estimates are known as Strichartz estimates. They are especially useful for the semilinear NLW without derivatives, and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces first appear in one-dimension in RaRe1982 and in higher dimensions in Be1983 in the context of propagation of singularities; they were used implicitly for LWP in KlMa1993, while the Schrodinger and KdV analogues were developed in Bo1993, Bo1993b.