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### Open Access Journals

CPAA

We consider the generalized Proudman-Johnson equation, in which an artificial parameter $a$ controlling the impact of convection was introduced to the Proudman-Johnson equation ([33]). In the present paper, we are going to show that there are global weak solutions to the generalized Proudman-Johnson equation for certain parameter $a$'s.

DCDS-B

We show local existence of solutions to a two-component Hunter--Saxton system. Moreover, we prove that the slopes of solutions can become unbounded. Finally, if initial data satisfy appropriate smallness conditions, the associated flow is global.

DCDS

Steady-states and traveling-waves of the generalized Constantin--Lax--Majda equation
are computed and their asymptotic behavior is described.
Their relation with possible blow-up and
the Benjamin--Ono equation is discussed.

CPAA

We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a

*weak Riemannian*structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.## Year of publication

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