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* LWP for <math>s \geq 1</math> by [[Strichartz estimate]]s (see e.g. [[LbSo1995]]; earlier references exist) ...e energy class [[Pl-p5]], [[FurPlTer2001]]. This argument extends to other energy-critical and sub-critical powers in dimensions 4 and higher.

** u <math> \in L^\infty_t L^2_x </math>(energy estimate) ...2_t (sharp Kato smoothing effect) [[KnPoVe1993]]. Earlier versions of this estimate were obtained in [[Ka1979b]], [[KrFa1983]].

* LWP for <math>s \geq 1/4</math> by [[Strichartz estimate]]s (see e.g. [[LbSo1995]]) ** For <math>s \geq 1</math> this is clear from energy conservation (for both NLKG and NLW).

** (Energy estimate) <math>f \in L^\infty_t L^2_x.</math> ...> [[Fc-p4]]. Similarly when <math>d=2\,</math> with the <math>L^4\,</math> estimate, which is also given by Gaussian beams with a constant of <math>2^{-1/2}\,.

...e]]s, for instance by exploiting the positivity properties of the [[stress-energy tensor]].

** (Energy estimate) If <math>f (0)</math> is in <math>H^s</math>, then <math>f (t)</math> is ** (Decay estimate) If <math>f (0)</math> has more than <math>(d+1)/2</math> derivatives in <

...2,\,</math> then one has blowup for the focusing nonlinearity for negative energy in the <math>L^2\,</math> supercritical or critical, <math>H^1\,</math> sub ...<math>p=1,\infty\,</math> and thus does not directly imply the dispersive estimate although it does give Strichartz estimates for <math>1 < p < \infty\,.</mat

...es standard techniques such as the energy method to fail (since the energy estimate does not recover any regularity in the case of the Schrodinger equation). H ...near the upper paraboloid, and these terms are more likely to disappear in energy estimates). One can often "gauge transform" the equation (in a manner depen

dispersive inequality and an energy inequality. ...able [[MacNkrNaOz2005]].However in the general case one cannot recover the estimate even if one uses the BMO norm or attempts Littlewood-Paley frequency locali

J. Bourgain, ''Scattering in the energy space and below for 3D NLS'', J. Anal. Math. '''75''' (1998), 267-297. N. Bournaveas, ''Local existence of energy class solutions for the Dirac-Klein-Gordon equations''. Comm. Partial Diffe