Abstract
We say that an evolution equation has an infinite gain of regularity if its solutions are C ∞ for t > 0, for initial data with only a finite amount of smoothness. An equation need not be hypoelliptic for this to happen provided the initial data vanish at spatial infinity. For instance, for the Schrödinger equation in R n, this is clear from the explicit solution formula if the initial data decay faster than any polynomial. For the Korteweg-deVries equation, T. Kato [4], motivated by work of A. Cohen, showed that the solutions are C ∞ for any data in L 2 with a weight function 1 + e σx . While the proof of Kato appears to depend on special a priori estimates, some of its mystery has been resolved by the recent results of finite regularity for various other nonlinear dispersive equations due to Constantin and Saut [1], Ponce [5] and others [3]. However, all of them require growth conditions on the nonlinear term.
Supported in part by NSF Grants DMS 87-22331, DMS 89-20624 and AFOSR Grant DAAL-3-86-0074. In addition, the first author is supported by an Alfred P. Sloan Foundation Fellowship.
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References
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© 1991 Springer-Verlag New York, Inc.
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Craig, W., Kappeler, T., Strauss, W. (1991). Infinite Gain of Regularity for Dispersive Evolution Equations. In: Beals, M., Melrose, R.B., Rauch, J. (eds) Microlocal Analysis and Nonlinear Waves. The IMA Volumes in Mathematics and its Applications, vol 30. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9136-4_5
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DOI: https://doi.org/10.1007/978-1-4613-9136-4_5
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