1. GLOBAL EXISTENCE AND LIMITING BEHAVIOR OF UNIDIRECTIONAL FLOCKS FOR THE FRACTIONAL EULER ALIGNMENT SYSTEM.
- Author
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LEAR, DANIEL
- Subjects
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ENTROPY - Abstract
In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels ϕ (x): = |x|(n+α) for α ∊ (0, 2). Here, we consider the critical case α = 1 and establish a couple of global existence results of smooth solutions, together with a description of their long time dynamics. The first one is obtained via Schauder-type estimates under a null initial entropy condition and the other is a small data result. We extend the notion of weak solution and prove the existence of global Leray--Hopf solutions for α ∊ [1, 2). Furthermore, we give anisotropic Onsager-type criteria for the validity of the natural energy law of the system. Finally, we provide quantitative estimates that show how far the density of the limiting flock is from a uniform distribution, depending solely on the size of the initial entropy. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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