Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A regularizing effect of nonlinear transport equations


Author: Felix Otto
Journal: Quart. Appl. Math. 56 (1998), 355-375
MSC: Primary 35L65; Secondary 35F25, 35Q35, 76S05, 82C70
DOI: https://doi.org/10.1090/qam/1622511
MathSciNet review: MR1622511
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Abstract: We consider the semigroup on $ {L^1}\left( {\mathbb{R}^n} \right)$ defined by the nonlinear transport equation for the scalar $ s$,

$\displaystyle {\partial _t}s + div\left( f\left( s \right)u \right) = 0 \qquad in \left( 0, \infty \right) \times {\mathbb{R}^n}$

for given velocity field $ u$. We show that this nonlinear semigroup is Hölder continuous for $ t > 0$ in the uniform operator topology, provided the graph of $ f$ has no linear segments. This continuity property--which expresses a regularizing effect of the nonlinearity in the transport equation--is robust with respect to the spatial behaviour of the time-independent velocity field $ u$.

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DOI: https://doi.org/10.1090/qam/1622511
Article copyright: © Copyright 1998 American Mathematical Society


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