Global Well-Posedness of 2D Euler-$α$ Equations in Exterior Domain

Abstract

After casting Euler-α equations into vorticity-stream function formula, we obtain some very useful estimates from the properties of the vorticity formula in exterior domain. Basing on these estimates, one can have got the global existence and uniqueness of the solutions to Euler-α equations in 2D exterior domain provided that the initial data is regular enough.

Global Well-Posedness of 2D Second Grade Fluid Equations in Exterior Domain

“…The difficult lies on that u n should be solenoidal and vanishes on boundary Γ. However, we have the following lemma, which can be found in [27].…”

“…There are many well known results about existence and a prior estimates, see [7] for example. To apply the method proposed in [27], we need the property that q is compactly supported in arbitrary finite time interval provided the initial data q 0 is compactly supported. However, from (1.6), formally we know that q(X(t, α), t) = q 0 (α)…”

“…However, we could not obtain (2.22) in general. We have the following lemma, which can be found in [27].…”

“…Solving this problem is not easy, the major difficult arises from the lack of 'good' a priori estimates on the derivatives of the solution, which is mainly due, on one hand, to the rather involved form of the nonlinearities and high order derivatives in (1.1), and on the other hand, to unboundedness of the domain. Our work here is inspired by the method used in [27] to solve well-posedness problem of Euler-α equations. By taking curl of equation (1.1), we consider instead the vorticity-stream formula…”

“…All in all, the main feature of the proof is based on constructing a sequence of approximated solutions by the method used in [27] and a limit process using compactness arguments in order to control the nonlinear terms. In addition, we will also make sense of the weak formulation introduced in the main theorem.…”