Abstract
In this article we study some problems related to the incompressible 3D Navier–Stokes equations from the point of view of Lebesgue spaces of variable exponent. These functional spaces present some particularities that make them quite different from the usual Lebesgue spaces: indeed, some of the most classical tools in analysis are not available in this framework. We will give here some ideas to overcome some of the difficulties that arise in this context in order to obtain different results related to the existence of mild solutions for this evolution problem.
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Notes
Recall that the Leray projector \(\mathbb {P}\) can also be defined in terms of the Riesz transforms: \(\mathbb {P}(\vec {\varphi })=(Id_{3\times 3}-\vec {R}\otimes \vec {R})(\vec {\varphi })\) where \(\vec {R}=(R_1, R_2, R_3)\) and \(R_j\) is the j-th Riesz transform. Thus, as long as the Riesz transforms are bounded in a functional space E, then the Leray projector \(\mathbb {P}\) is bounded in E and we have \(\Vert \mathbb {P}(\vec {\varphi })\Vert _E\le C\Vert \vec {\varphi }\Vert _E\).
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Gastón Vergara-Hermosilla is supported by the ANID postdoctoral program BCH 2022 Grant No. 74220003.
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DCh and GV-H wrote the main manuscript text. All authors reviewed the manuscript.
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Chamorro, D., Vergara-Hermosilla, G. Lebesgue spaces with variable exponent: some applications to the Navier–Stokes equations. Positivity 28, 24 (2024). https://doi.org/10.1007/s11117-024-01043-6
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DOI: https://doi.org/10.1007/s11117-024-01043-6