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Random initial conditions for semi-linear PDEs

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  29 January 2019

Dirk Blömker ,
Giuseppe Cannizzaro  and
Marco Romito
Dirk Blömker
Affiliation:
Institüt fur Mathematik, Universität Augsburg, D-86135 Augsburg, Germany (dirk.bloemker@math.uni-augsburg.de)
Giuseppe Cannizzaro
Affiliation:
Imperial College London, Department of Mathematics, 180 Queen's Gate, LondonSW7 2AZ, UK (g.cannizzaro@imperial.ac.uk)
Marco Romito
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I–56127Pisa, Italia (marco.romito@unipi.it); URL: http://people.dm.unipi.it/romito
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Abstract

We analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.

In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.

We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.

The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

Keywords

Random initial conditionsemilinar PDEs

MSC classification

Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1533 - 1565
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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