Non-stationary flow of an ideal incompressible liquid

Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

“…In order to show the uniqueness part of our statement, we shall use the Yudovich argument [17] revisited by P. Gérard in [11]. Let (θ 1 , u 1 , Π 1 ) and (θ 2 , u 2 , Π 2 ) satisfy (2) and (B κ,0 ) with the same data.…”

“…The present paper aims at extending the celebrated result by Yudovich concerning the two-dimensional Euler system (see [17]) to the following two-dimensional Boussinesq system:…”

Large Time Behavior for Vortex Evolution in the Half-Plane

“…If ω 0 is bounded then (1.1) is globally well-posed since it is equivalent, through the method of images, to an initial-value problem in the full-plane, with bounded, compactly supported initial vorticity (shown to be wellposed by Yudovich in [37]). The method of images consists of the observation that the Euler equations are covariant with respect to mirror-symmetry.…”

“…For bounded, compactly supported initial vorticity, problem (1.1) is equivalent to the full plane problem with initial vorticity given by an odd extension of ω 0 to {x 2 < 0} (see [17] for details). Global well-posedness of the initial boundary value problem follows from this equivalence, using Yudovich's Theorem [37]. For compactly supported initial vorticity in (L 1 + BM + ) ∩ H −1 loc we have global existence of weak solutions by adapting Delort's Theorem to the half-plane case, see [4,17,33,35], but uniqueness is open.…”

Local Regularity for the Modified SQG Patch Equation

“…The latter, including hurricanes and tornados, are common in nature. Existence and uniqueness of vortex patch solutions to the 2D Euler equation on the whole plane goes back to the work of Yudovich [30], and regularity in this setting refers to a sufficient smoothness of the patch boundaries as well as to a lack of both self-intersections of each patch boundary and touching of different patches.…”