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On a Non-Periodic Modified Euler Equation: Well-Posedness and Quasi-Invariant Measures

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We consider a modified Euler equation on \(\mathbb {R}^{2}\). We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are quasi-invariant for the Euler flow. Almost everywhere with respect to such measures (and, in particular, for less regular initial conditions), the flow is shown to be globally defined.

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The authors thank Prof. Nikolay Tzvetkov for very useful discussions. They acknowledge the support of FCT project PTDC/MAT-STA/0975/2014. The second author was also funded by the LisMath fellowship PD/BD/52641/2014, FCT, Portugal.

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Correspondence to Alexandra Symeonides.

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Cruzeiro, A.B., Symeonides, A. On a Non-Periodic Modified Euler Equation: Well-Posedness and Quasi-Invariant Measures. Potential Anal 54, 607–626 (2021).

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Mathematics Subject Classification 2010