Wave estimates: Difference between revisions
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Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms | Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms $L^q_t L^r_x$, or in $X^{s,b}$ spaces, defined by | ||
$$\| u \|_{X^{s,b}} = \| \langle\xi\rangle^s \langle|\xi| - |\tau|\rangle^b \hat{u} ( \tau, \xi )\|_2 $$ | |||
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 $X^{s,b}$ 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 | |||
== Specific wave estimates == | == Specific wave estimates == | ||
* [[Linear wave estimates]] ([[Strichartz estimates|Strichartz]], etc.) | * [[Linear wave estimates]] ([[Strichartz estimates|Strichartz]], [[Morawetz-type estimates|Morawetz-KSS]], [[Weighted Strichartz estimates|Weighted Strichartz]], etc.) | ||
* [[Bilinear wave estimates]] | * [[Bilinear wave estimates]] | ||
* [[Multilinear wave estimates]] | * [[Multilinear wave estimates]] | ||
Latest revision as of 22:28, 8 March 2011
Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms $L^q_t L^r_x$, or in $X^{s,b}$ spaces, defined by $$\| u \|_{X^{s,b}} = \| \langle\xi\rangle^s \langle|\xi| - |\tau|\rangle^b \hat{u} ( \tau, \xi )\|_2 $$
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 $X^{s,b}$ 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.
