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Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data

J. F. was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. J. F. also acknowledges support from Tadahiro Oh's ERC starting grant no. 637995 "ProbDynDispEq"

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  • We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus $ \mathbb{T} $ with initial data below $ L^{2}( \mathbb{T}) $. With respect to random initial data of strictly negative Sobolev regularity, we prove that BBM is almost surely globally well-posed. The argument employs the $ I $-method to obtain an a priori bound on the growth of the 'residual' part of the solution. We then discuss the stability properties of the solution map in the deterministically ill-posed regime.

    Mathematics Subject Classification: Primary: 35Q53, 76B15.


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