Finite time extinction for the 1D stochastic porous medium equation with transport noise

Hensel S. 2021. Finite time extinction for the 1D stochastic porous medium equation with transport noise. Stochastics and Partial Differential Equations: Analysis and Computations.

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Abstract
We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate O(t12) whereas the support of solutions to the deterministic PME grows only with rate O(t1m+1). The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.
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Date Published
2021-03-21
Journal Title
Stochastics and Partial Differential Equations: Analysis and Computations
Acknowledgement
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385 . I am very grateful to M. Gerencsér and J. Maas for proposing this problem as well as helpful discussions. Special thanks go to F. Cornalba for suggesting the additional κ-truncation in Proposition 5. I am also indebted to an anonymous referee for pointing out a gap in a previous version of the proof of Lemma 9 (concerning the treatment of the noise term). The issue is resolved in this version.
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Hensel S. Finite time extinction for the 1D stochastic porous medium equation with transport noise. Stochastics and Partial Differential Equations: Analysis and Computations. 2021. doi:10.1007/s40072-021-00188-9
Hensel, S. (2021). Finite time extinction for the 1D stochastic porous medium equation with transport noise. Stochastics and Partial Differential Equations: Analysis and Computations. Springer Nature. https://doi.org/10.1007/s40072-021-00188-9
Hensel, Sebastian. “Finite Time Extinction for the 1D Stochastic Porous Medium Equation with Transport Noise.” Stochastics and Partial Differential Equations: Analysis and Computations. Springer Nature, 2021. https://doi.org/10.1007/s40072-021-00188-9.
S. Hensel, “Finite time extinction for the 1D stochastic porous medium equation with transport noise,” Stochastics and Partial Differential Equations: Analysis and Computations. Springer Nature, 2021.
Hensel S. 2021. Finite time extinction for the 1D stochastic porous medium equation with transport noise. Stochastics and Partial Differential Equations: Analysis and Computations.
Hensel, Sebastian. “Finite Time Extinction for the 1D Stochastic Porous Medium Equation with Transport Noise.” Stochastics and Partial Differential Equations: Analysis and Computations, Springer Nature, 2021, doi:10.1007/s40072-021-00188-9.
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2021-04-06
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