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Free keywords:
Euler equation; Singularities
Abstract:
We present a study of complex singularities of a two-parameter family of solutions for the two-dimensional Euler equation with periodic boundary conditions and initial conditions View the MathML sourceψ0(z1,z2)=Fˆ(p)cosp⋅z+Fˆ(q)cosq⋅z in the short-time asymptotic régime. As has been shown numerically in Pauls et al. [W. Pauls, T. Matsumoto, U. Frisch, J. Bec, Nature of complex singularities for the 2D Euler equation, Physica D 219 (2006) 40–59], the type of the singularities depends on the angle ϕϕ between the modes pp and qq. Thus, the Fourier coefficients of the solutions decrease as View the MathML sourceG(k,θ)∼C(θ)k−αe−δ(θ)k with the exponent αα depending on ϕϕ. Here we show for the two particular cases of ϕϕ going to zero and to ππ that the type of the singularities can be determined very accurately, being characterised by α=5/2α=5/2 and α=3α=3 respectively. In these two cases we are also able to determine the subdominant corrections. Furthermore, we find that the geometry of the singularities in these two cases is completely different, the singular manifold being located “over” different points in the real domain.
