Ill-posedness of the 3D incompressible hyperdissipative Navier–Stokes system in critical Fourier-Herz spaces

Abstract

We study the Cauchy problem for the 3D incompressible hyperdissipative Navier-Stokes equations and consider the well-posedness and ill-posedness. We prove that if p > 6 8−α and q 1, the system is locally well-posed for large initial data as well as globally wellposed for small initial data. Also, we obtain the same result for p = 6 8−α and q ∈ [ 6 8−α , 2]. More importantly, we show that the system is ill-posed in the sense of norm inflation for p = 6 8−α and q > 2. The proof relies heavily on particular structure of initial data u 0 that we construct, which makes the first iteration of solution inflate. Specifically, the special structure of u 0 transforms an infinite sum into a finite sum in 'remainder term', which permits us to control the remainder.

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A new Gammarus species from Xinjiang Uygur Autonomous Region (China) with a key to Xinjiang freshwater gammarids (Crustacea, Amphipoda, Gammaridae)

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