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Sharp Global Well-posedness and Scattering of the Boltzmann Equation
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abstract
We consider the 3D Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff. We prove sharp global well-posedness with initial data small in the scaling-critical space. The solution also remains in $L^{1}$ if the initial datum is in $L^{1}$, even at such low regularity. The key to existence, uniqueness and regularity criteria is the new bilinear spacetime estimates for the gain term, the proof of which is based on novel techniques from nonlinear dispersive PDEs including the atomic $U$-$V$ spaces, multi-linear frequency analysis, dispersive estimates, etc. To our knowledge, this is the first 3D sharp global result for the Boltzmann equation.
Forward citations
Cited by 2 Pith papers
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Scaling-Critical Theory for the Boltzmann and Landau Equations
Global well-posedness near Maxwellian for Landau and very soft non-cutoff Boltzmann equations in a new scaling-critical norm, with pointwise Green function estimates.
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Global well-posedness of the Boltzmann equation via bilinear estimates
Global well-posedness holds for the hard-sphere Boltzmann equation with small data in the critical spaces B^{d-1}_{1,1} L_v^1 and W^{d-1,1} L_v^1 via new transversality bilinear estimates.
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