Solutions of Navier–Stokes–Maxwell systems in large energy spaces
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- by Diogo Arsénio and Isabelle Gallagher PDF
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Abstract:
Large weak solutions to Navier–Stokes–Maxwell systems are not known to exist in their corresponding energy space in full generality. Here, we mainly focus on the three-dimensional setting of a classical incompressible Navier–Stokes–Maxwell system and—in an effort to build solutions in the largest possible functional spaces—prove that global solutions exist under the assumption that the initial velocity and electromagnetic fields have finite energy, and that the initial electromagnetic field is small in $\dot H^s\left ({\mathbb R}^3\right )$ with $s\in \left [\frac 12,\frac 32\right )$. We also apply our method to improve known results in two dimensions by providing uniform estimates as the speed of light tends to infinity.
The method of proof relies on refined energy estimates and a Grönwall-like argument, along with a new maximal estimate on the heat flow in Besov spaces. The latter parabolic estimate allows us to bypass the use of the so-called Chemin–Lerner spaces altogether, which is crucial and could be of independent interest.
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Additional Information
- Diogo Arsénio
- Affiliation: Department of Mathematics, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates
- Email: diogo.arsenio@nyu.edu
- Isabelle Gallagher
- Affiliation: DMA, École normale supérieure, CNRS, PSL Research University, 75005 Paris; and UFR de mathématiques, Université Paris-Diderot, Sorbonne Paris-Cité, 75013 Paris, France
- MR Author ID: 617258
- Email: gallagher@math.ens.fr
- Received by editor(s): November 4, 2018
- Published electronically: March 9, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 3853-3884
- MSC (2010): Primary 35Q35
- DOI: https://doi.org/10.1090/tran/8000
- MathSciNet review: 4105512