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Abstract We investigate global solutions to the Euler-alignment system in $$d$$ dimensions with unidirectional flows and strongly singular communication protocols $$\phi (x) = |x|^{-(d+\alpha )}$$ for $$\alpha \in (0,2)$$. Our paper establishes global regularity results in both the subcritical regime $$1<\alpha <2$$ and the critical regime $$\alpha =1$$. Notably, when $$\alpha =1$$, the system exhibits a critical scaling similar to the critical quasi-geostrophic equation. To achieve global well-posedness, we employ a novel method based on propagating the modulus of continuity. Our approach introduces the concept of simultaneously propagating multiple moduli of continuity, which allows us to effectively handle the system of two equations with critical scaling. Additionally, we improve the regularity criteria for solutions to this system in the supercritical regime $$0<\alpha <1$$.
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Abstract In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behaviour and optimal decay estimates of the solutions as .more » « less
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