Abstract
We prove the pointwise decay of solutions to three linear equations: (1) the transport equation in phase space generalizing the classical Vlasov equation, (2) the linear Schrödinger equation, (3) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain \(L^1\)—\(L^\infty \) decay through directly estimating the fundamental solution in physical space or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines “vector field” commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space–time weighted energy estimates. Combined with a simple version of Sobolev’s inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrödinger equations, we can recover sharp pointwise decay; in the Schrödinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on \(L^2\)—\(L^\infty \) decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.
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References
J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer, New York, 1976.
S. Chen and Y. Zhou, Decay rate of solutions to hyperbolic system of first order, Acta Math. Sin. (Engl. Ser.) 15 (1999), 471–484.
R. Donninger and J. Krieger, A vector field on the distorted Fourier side and decay for wave equations with potentials, Mem. Amer. Math. Soc. 241 (2016), no. 1142.
D. Fajman, J. Joudioux, and J. Smulevici, A vector field method for relativistic transport equations with applications, Anal. PDE 10 (2017), 1539–1612.
M. Keel and T. Tao, Endpoint Strichartz estimate, Amer. J. Math. 120 (1998), 955–980.
S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Commun. Pure Appl. Math. 38 (1985), 321–332.
S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein–Gordon equations in four space–time dimensions, Commun. Pure Appl. Math. 38 (1985), 631–641.
S. Klainerman, A commuting vectorfields approach to Strichartz-type inequalities and applications to quasilinear wave equations, Int. Math. Res. Not. IMRN 2001 (2001), 221–274.
S. Klainerman, I. Rodnianski, and T. Tao, A physical space approach to wave equation bilinear estimates, J. Anal. Math. 87 (2002), 299–336. Dedicated to the memory of Thomas H. Wolff.
P. G. LeFloch and Y. Ma, The Hyperboloidal Foliation Method, Series in Applied and Computational Mathematics, vol. 2, World Scientific, Hackensack, NJ, 2015.
C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis, Volume 1, Cambridge University Press, New York, 2013.
F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 261–290 (English, with English and French summaries).
J. Smulevici, Small data solutions of the Vlasov–Poisson system and the vector field method, Ann. PDE 2 (2016) Art. 11, 55 pp.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1995.
E. M. Stein and R. Shakarchi, Functional Analysis, Introduction to Further Topics in Analysis, Princeton Lectures in Analysis, vol. 4, Princeton University Press, Princeton, NJ, 2011.
T. Tao, A physical space proof of the bilinear Strichartz and local smoothing estimates for the Schrödinger equation (2013), unpublished, available at https://terrytao.wordpress.com/2013/07/10/.
Q. Wang, An intrinsic hyperboloid approach for Einstein Klein–Gordon equations (2016), preprint, available at arXiv:1607.01466.
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Wong, W.W.Y. A commuting-vector-field approach to some dispersive estimates. Arch. Math. 110, 273–289 (2018). https://doi.org/10.1007/s00013-017-1114-4
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DOI: https://doi.org/10.1007/s00013-017-1114-4