Difference between revisions of "Linear Airy estimates"

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* If <math>u \in X^{0,1/2+}</math> on '''R''', then
 
* If <math>u \in X^{0,1/2+}</math> on '''R''', then
 
** u <math> \in L^\infty_t L^2_x </math>(energy estimate)
 
** u <math> \in L^\infty_t L^2_x </math>(energy estimate)
** <span class="SpellE">D_x</span>^{1/4} u is in L^4_t <span class="SpellE">BMO_x</span> (endpoint <span class="SpellE">Strichartz</span>) [[references.html#KnPoVe1993 KnPoVe1993]]
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** <span class="SpellE">D_x</span>^{1/4} u is in L^4_t <span class="SpellE">BMO_x</span> (endpoint <span class="SpellE">Strichartz</span>) [[Bibliography#KnPoVe1993|KnPoVe1993]]
** <span class="SpellE">D_x</span> u is in L^\<span class="SpellE">infty_x</span> L^2_t (sharp Kato smoothing effect) [[references.html#KnPoVe1993 KnPoVe1993]]. Earlier versions of this estimate were obtained in [[references.html#Ka1979b Ka1979b]], [[references.html#KrFa1983 KrFa1983]].
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** <span class="SpellE">D_x</span> u is in L^\<span class="SpellE">infty_x</span> L^2_t (sharp Kato smoothing effect) [[Bibliography#KnPoVe1993|KnPoVe1993]]. Earlier versions of this estimate were obtained in [[Bibliography#Ka1979b|Ka1979b]], [[Bibliography#KrFa1983|KrFa1983]].
** <span class="SpellE">D_x</span>^{-1/4} u is in L^4_x L^\<span class="SpellE">infty_t</span> (Maximal function) [[references.html#KnPoVe1993 KnPoVe1993]], [[references.html#KnRu1983 KnRu1983]]
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** <span class="SpellE">D_x</span>^{-1/4} u is in L^4_x L^\<span class="SpellE">infty_t</span> (Maximal function) [[Bibliography#KnPoVe1993|KnPoVe1993]], [[Bibliography#KnRu1983|KnRu1983]]
** <span class="SpellE">D_x</span>^{-3/4-} u is in L^2_x L^\<span class="SpellE">infty_t</span> (L^2 maximal function) [[references.html#KnPoVe1993 KnPoVe1993]]
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** <span class="SpellE">D_x</span>^{-3/4-} u is in L^2_x L^\<span class="SpellE">infty_t</span> (L^2 maximal function) [[Bibliography#KnPoVe1993|KnPoVe1993]]
 
** ''Remark''<nowiki>: Further estimates are available by </nowiki><span class="SpellE">Sobolev</span>, differentiation, Holder, and interpolation. For instance:
 
** ''Remark''<nowiki>: Further estimates are available by </nowiki><span class="SpellE">Sobolev</span>, differentiation, Holder, and interpolation. For instance:
*** <span class="SpellE">D_x</span> u is in L^2_{<span class="SpellE">x,t</span>} locally in space [[references.html#Ka1979b Ka1979b]] - use Kato and Holder (can also be proven directly by integration by parts)
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*** <span class="SpellE">D_x</span> u is in L^2_{<span class="SpellE">x,t</span>} locally in space [[Bibliography#Ka1979b|Ka1979b]] - use Kato and Holder (can also be proven directly by integration by parts)
 
*** u is in L^2_{<span class="SpellE">x,t</span>} locally in time - use energy and Holder
 
*** u is in L^2_{<span class="SpellE">x,t</span>} locally in time - use energy and Holder
 
*** <span class="SpellE">D_x</span>^{3/4-} u is in L^8_x L^2_t locally in time - interpolate previous with Kato
 
*** <span class="SpellE">D_x</span>^{3/4-} u is in L^8_x L^2_t locally in time - interpolate previous with Kato
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*** <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is in L^8_t L^4_x - interpolate energy with endpoint <span class="SpellE">Strichartz</span>. (In particular, <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is also in L^4_{<span class="SpellE">x,t</span>}).
 
*** <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is in L^8_t L^4_x - interpolate energy with endpoint <span class="SpellE">Strichartz</span>. (In particular, <span class="SpellE">D_x</span><span class="GramE">^{</span>1/8} u is also in L^4_{<span class="SpellE">x,t</span>}).
 
*** u is in L^8_{<span class="SpellE">x,t</span>}- use previous and <span class="SpellE">Sobolev</span> in space
 
*** u is in L^8_{<span class="SpellE">x,t</span>}- use previous and <span class="SpellE">Sobolev</span> in space
*** If u is in X^{0,1/3+}, then u is in L^4_{<span class="SpellE">x,t</span>} [[references.html#Bo1993b Bo1993b]] - interpolate previous with the trivial identity X^{0,0} = L^2
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*** If u is in X^{0,1/3+}, then u is in L^4_{<span class="SpellE">x,t</span>} [[Bibliography#Bo1993b|Bo1993b]] - interpolate previous with the trivial identity X^{0,0} = L^2
*** If u is in X^{0,1/4+}, then <span class="SpellE">D_x</span>^{1/2} u is in L^4_x L^2_t [[references.html#Bo1993b Bo1993b]] - interpolate Kato with X^{0,0} = L^2
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*** If u is in X^{0,1/4+}, then <span class="SpellE">D_x</span>^{1/2} u is in L^4_x L^2_t [[Bibliography#Bo1993b|Bo1993b]] - interpolate Kato with X^{0,0} = L^2
 
* If u is in X^{0,1/2+} on '''T''', then
 
* If u is in X^{0,1/2+} on '''T''', then
 
** <span class="GramE">u</span> is in L^\<span class="SpellE">infty_t</span> L^2_x (energy estimate). This is also true in the large period case.
 
** <span class="GramE">u</span> is in L^\<span class="SpellE">infty_t</span> L^2_x (energy estimate). This is also true in the large period case.
** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time (in fact one only needs u in X^{0,1/3} for this) [[references.html#Bo1993b Bo1993b]].
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** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time (in fact one only needs u in X^{0,1/3} for this) [[Bibliography#Bo1993b|Bo1993b]].
** <span class="SpellE">D_x</span><span class="GramE">^{</span>-\<span class="SpellE">eps</span>} u is in L^6_{<span class="SpellE">x,t</span>} locally in time. [[references.html#Bo1993b Bo1993b]]. It is conjectured that this can be improved to L^8_{<span class="SpellE">x<span class="GramE">,t</span></span>}.
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** <span class="SpellE">D_x</span><span class="GramE">^{</span>-\<span class="SpellE">eps</span>} u is in L^6_{<span class="SpellE">x,t</span>} locally in time. [[Bibliography#Bo1993b|Bo1993b]]. It is conjectured that this can be improved to L^8_{<span class="SpellE">x<span class="GramE">,t</span></span>}.
 
** ''Remark''<nowiki>: there is no smoothing on the circle, so one can never gain regularity.</nowiki>
 
** ''Remark''<nowiki>: there is no smoothing on the circle, so one can never gain regularity.</nowiki>
 
* If u is in X^{0,1/2} on a circle with large period \lambda, then
 
* If u is in X^{0,1/2} on a circle with large period \lambda, then
 
** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time, with a bound of \lambda^{0+}.
 
** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time, with a bound of \lambda^{0+}.
*** 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]]\
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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. [[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]].
  
 
[[Category:Airy]]
 
[[Category:Airy]]
 
[[Category:Estimates]]
 
[[Category:Estimates]]

Revision as of 16:25, 31 July 2006

The following linear estimates for the Airy equation are known:

  • If on R, then
    • u (energy estimate)
    • D_x^{1/4} u is in L^4_t BMO_x (endpoint Strichartz) KnPoVe1993
    • D_x u is in L^\infty_x L^2_t (sharp Kato smoothing effect) KnPoVe1993. Earlier versions of this estimate were obtained in Ka1979b, KrFa1983.
    • D_x^{-1/4} u is in L^4_x L^\infty_t (Maximal function) KnPoVe1993, KnRu1983
    • D_x^{-3/4-} u is in L^2_x L^\infty_t (L^2 maximal function) KnPoVe1993
    • Remark: Further estimates are available by Sobolev, differentiation, Holder, and interpolation. For instance:
      • D_x u is in L^2_{x,t} locally in space Ka1979b - use Kato and Holder (can also be proven directly by integration by parts)
      • u is in L^2_{x,t} locally in time - use energy and Holder
      • D_x^{3/4-} u is in L^8_x L^2_t locally in time - interpolate previous with Kato
      • D_x^{1/6} u is in L^6_{x,t} - interpolate energy with endpoint Strichartz (or Kato with maximal)
      • D_x^{1/8} u is in L^8_t L^4_x - interpolate energy with endpoint Strichartz. (In particular, D_x^{1/8} u is also in L^4_{x,t}).
      • u is in L^8_{x,t}- use previous and Sobolev in space
      • If u is in X^{0,1/3+}, then u is in L^4_{x,t} Bo1993b - interpolate previous with the trivial identity X^{0,0} = L^2
      • If u is in X^{0,1/4+}, then D_x^{1/2} u is in L^4_x L^2_t Bo1993b - interpolate Kato with X^{0,0} = L^2
  • 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) Bo1993b.
    • D_x^{-\eps} u is in L^6_{x,t} locally in time. 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. CoKeStTkTa-p2.