Linear Airy estimates: Difference between revisions

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*** In fact, when u has frequency N, the constant is like \lambda^{0+} (N<span class="GramE">^{</span>-1/8} + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line in L^4 mentioned earlier. [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]\
*** In fact, when u has frequency N, the constant is like \lambda^{0+} (N<span class="GramE">^{</span>-1/8} + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line in L^4 mentioned earlier. [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]\


[[Category:Airy]]
[[Category:Estimates]]
[[Category:Estimates]]

Revision as of 05:03, 29 July 2006

The following linear estimates for the Airy equation are known:

  • If on R, then
  • If u is in X^{0,1/2+} on T, then
    • u is in L^\infty_t L^2_x (energy estimate). This is also true in the large period case.
    • u is in L^4_{x,t} locally in time (in fact one only needs u in X^{0,1/3} for this) references.html#Bo1993b Bo1993b.
    • D_x^{-\eps} u is in L^6_{x,t} locally in time. references.html#Bo1993b Bo1993b. It is conjectured that this can be improved to L^8_{x,t}.
    • Remark: there is no smoothing on the circle, so one can never gain regularity.
  • If u is in X^{0,1/2} on a circle with large period \lambda, then
    • u is in L^4_{x,t} locally in time, with a bound of \lambda^{0+}.
      • In fact, when u has frequency N, the constant is like \lambda^{0+} (N^{-1/8} + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line in L^4 mentioned earlier. references.html#CoKeStTaTk-p2 CoKeStTkTa-p2\