Abstract
We study the smoothing effects of diffusion in a 3D vector system of linear advection-diffusion PDEs known as the kinematic dynamo equations. We examine the case where both the divergence free drift velocity \({\mathbf{u}(x,t)}\) and the mild solution \({\mathbf{B}(x,t)}\) to the PDE obey the natural scaling of the equation. We consider initial data \({\mathbf{B}_0(x) \in L^3(\mathbb{R}^3)}\) and demonstrate how the regularization effects of diffusion are influenced by the profile of \({\mathbf{u}(x,t)}\).
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Friedlander, S., Rusin, W. On the Smoothing Effect in the Kinematic Dynamo Equations in Critical Spaces. J. Math. Fluid Mech. 17, 145–153 (2015). https://doi.org/10.1007/s00021-014-0199-9
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DOI: https://doi.org/10.1007/s00021-014-0199-9