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Forbes contributors publish independent expert analyses and insights. I write about the people, business and politics of the wine industry.

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small H s H s -initial conditions (with s s sufficiently large), in solutions to ...

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Instead of regularizing f (ρ) = lnρ f (ρ) = ln ρ in the LogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE by first regularizing ...