Finite time blow-up for the hypodissipative Navier Stokes equations with a force in L¹_t C_x^1,ε∩ L^∞_tL_x²

2024/07/09 by Córdoba, Diego, Martínez-Zoroa, Luis, Zheng, Fan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.06776

Abstract

In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by |∇|^α for any α∈ [0, α₀) (α₀ = (22-8√7)/(9) > 0). We construct solutions in ℝ³× [0,T] with a finite T>0 and with an external forcing which is in L¹_t([0, T]) C_x^1,ε∩ L^∞_tL_x², such that on the time interval 0 ≤ t < T, the velocity u is in the space C^∞∩ L² and such that as the time t approaches the blow-up moment T, the integral ∫₀^t |∇ u| ds tends to infinity.

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