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Recent Progress in Smoothing Estimates for Evolution Equations

  • Michael Ruzhansky
  • Mitsuru SugimotoEmail author
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)

Abstract

This paper is a survey article of results and arguments from authors’ papers (Ruzhansky and Sugimoto in Proc. Lond. Math. Soc. 105:393–423, 2012; Ruzhansky and Sugimoto in Smoothing properties of non-dispersive equations; Ruzhansky and Sugimoto in Smoothing properties of inhomogeneous equations via canonical transforms), and describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on ideas of comparison principle and canonical transforms. For operators a(D x ) of order m satisfying the dispersiveness condition ∇a(ξ)≠0, the smoothing estimate
$$ {\bigl \Vert {{ \langle{x} \rangle}^{-s}|D_x|^{(m-1)/2}e^{ita(D_x)} \varphi(x)}\bigr \Vert }_{L^2{ ({{\mathbb{R}}_t\times {\mathbb{R}}^n_x} )}} \leq C{\Vert {\varphi} \Vert }_{L^2{ ({{\mathbb{R}}^n_x} )}} \quad(s>1/2) $$
is established, while it is known to fail for general non-dispersive operators. Especially, time-global smoothing estimates for the operator a(D x ) with lower order terms are the benefit of our new method. For the case when the dispersiveness breaks, we suggest a form
$$ {\bigl \Vert {{ \langle{x} \rangle}^{-s}\bigl|\nabla a(D_x)\bigr|^{1/2} e^{it a(D_x)}\varphi(x)}\bigr \Vert }_{L^2{ ({{\mathbb{R}}_t\times {\mathbb{R}}^n_x} )}} \leq C{\Vert {\varphi} \Vert }_{L^2{ ({{\mathbb{R}}^n_x} )}} \quad(s>1/2) $$
which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a(D x ). It does continue to hold for a variety of non-dispersive operators a(D x ), where ∇a(ξ) may become zero on some set. It is remarkable that our method allows us to carry out a global microlocal reduction of equations to the translation invariance property of the Lebesgue measure.

Keywords

Smoothing estimates Dispersive equations 

Mathematics Subject Classification (2010)

35B65 35Q35 35Q40 

Notes

Acknowledgements

The authors were supported by the Daiwa Anglo-Japanese Foundation. The first author was also supported by the EPSRC Leadership Fellowship EP/G007233/1.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Graduate School of MathematicsNagoya UniversityNagoyaJapan

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