2016/02/02 by Itoh, Tsubasa, Miura, Hideyuki, Yoneda, Tsuyoshi
#35Q31 #76B03 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1602.00815
We consider the two-dimensional Euler equations in non-smooth domains with corners. It is shown that if the angle of the corner θ is strictly less than π/2, the Lipschitz estimate of the vorticity at the corner is at most single exponential growth and the upper bound is sharp. %near the stagnation point. For the corner with the larger angle π/2 < θ<2π, θ≠ π, we construct an example of the vorticity which loses continuity instantaneously. For the case θ≤ π/2, the vorticity remains continuous inside the domain. We thus identify the threshold of the angle for the vorticity maintaining the continuity. For the borderline angle θ=π/2, it is also shown that the growth rate of the Lipschitz constant of the vorticity can be double exponential, which is the same as in Kiselev-Sverak's result (Annals of Math., 2014).