Self-Similar Solutions of the Two-Dimensional Incompressible Euler Equation from Large Initial Data

2026/07/22 by Hyungjun Choi
#math.AP

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Abstract

Let (1)/(3)<a<1 and let u₀ be a C¹, divergence-free, (-a)-homogeneous vector field on ℝ²∖\0\. We construct a forward-in-time self-similar solution of the two-dimensional incompressible Euler equations, u(t,x)=t^-(a)/(1+a) U(\fracxt^(1)/(1+a)), with initial datum u₀. No smallness or sign assumption is imposed on the initial datum. The construction is a vanishing-dissipation limit of the hypodissipative self-similar profiles. The key is a vorticity profile estimate, uniform in the dissipation parameter, in the critical Lorentz space L^{(2)/(1+a),∞}(ℝ²). The resulting Euler solution u has locally finite energy and ∇ u ∈ L^∞_t L^{(2)/(1+a),∞}_x. Indeed, the velocity is continuous in L²_loc, and the vorticity converges weak-* in L^{(2)/(1+a),∞} at the initial time.

Citations

• On putative self-similarity for incompressible 3D Euler
• Asymmetric Self-similar Spiral Solutions of 2-D Incomressible Euler Equations

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