A BKM-type criterion for the Euler equations

2025/05/23 by Aydın, Mustafa Sencer
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.18304

Abstract

We establish a new BKM-type blow-up criterion for solutions of the incompressible Euler equations that belong to Sobolev or H" older spaces. Our criterion involves the L² norm in time of the L^∞ norm of the first order tangential derivatives. Moreover, it applies to various domains such as the full space, the half-space, torus, (in)finite channel, and domains with curved boundaries. Additionally, we provide a mixed criterion involving the L¹_t L^∞(Ω₁) norm of the vorticity and the L²_t L^∞(Ω₂) norm of the first order conormal derivatives of the velocity where Ω₁ ∪ Ω₂ = Ω is a suitable decomposition of the physical space. Finally, we prove a blow-up criterion for the class of solutions that belong to the Sobolev conormal spaces that is recently constructed in~\citeAK1.

Citations

• Inviscid limit on L^p-based Sobolev conormal spaces for the 3D Navier-Stokes equations with the Navier boundary conditions
• Euler Equations in Sobolev conormal spaces
• Uniform bounds and the inviscid limit for the Navier-Stokes equations with Navier boundary conditions
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• Uniform time of existence for the alpha Euler equations
• Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations
• Geometric properties and non-blowup of 3-D incompressible Euler flow
• The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations
• An Eulerian-Lagrangian Approach for Incompressible Fluids: Local Theory

Related

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• Geometric blow-up criteria for the non-homogeneous incompressible Euler equations in 2-D
• Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension
• The Euler equations in a critical case of the generalized Campanato space
• Ill-posedness for the incompressible Euler equations in critical Sobolev spaces