Strichartz estimates: Difference between revisions

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Strichartz estimates are spacetime estimates on homogeneous and
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inhomogeneous linear dispersive and wave equations. They are
  ==
particularly useful for solving semilinear perturbations of such
equations, in which no derivatives are present in the nonlinearity.


Strichartz estimates can be derived abstractly as a consequence of a
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dispersive inequality and an energy inequality.


==Linear Strichartz estimate ==
Let <math> \dot H^{\alpha}(\Bbb R^n) </math> denote the homogeneous
Sobolev space with norm
<center>
<math>
\left\| u \right\|_{\dot H^{\alpha}(\Bbb R^n)}
= \left\|(-\Delta^{\alpha/2}) u \right\|_{L^2(\Bbb R^n)}
</math>
</center>
If <math> u </math> solves the ''linear wave equation''
<center>
<math>
\Box u = F(t,x)
</math>
</center>
with data
<center>
<math>
u(0,\cdot)=f \qquad \partial_t u (0,\cdot )=g
</math>
</center>
then the Strichartz estimates states that


<center> <math>
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\left\| u \right\|_{L^{4}({{\Bbb R}^{3+1}}_+)}
\leq C \left(
\left\| f \right\|_{{\dot H^{1/2}}(\Bbb R^3)}
+ \left\| g \right\|_{{\dot H^{-1/2}}(\Bbb R^3)}
+ \int\limits_0^{\infty} \left\| F \right\|_{L^2(\Bbb R^{3})}
\right)
</math> </center>
 
 
[[Category:Estimates]] [[Category:Schrodinger]]  [[Category:Wave]]
[[Category:Airy]]

Revision as of 19:27, 26 January 2011

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