In this paper we study the motion of a surface gravity wave with viscosity. In particular we prove two well-posedness results. On the one hand, we establish the local solvability in Sobolev spaces for arbitrary dissipation. On the other hand, we establish the global well-posedness in Wiener spaces for a sufficiently large viscosity. These results are the first rigorous proofs of well-posedness for the Dias, Dyachenko \& Zakharov system ({\em Physics Letters A} 2008) modelling gravity waves with viscosity.

中文翻译：

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具有粘性问题的水波的适定性

在本文中，我们研究了具有粘性的表面重力波的运动。特别地，我们证明了两个适定性结果。一方面，我们在 Sobolev 空间中建立了任意耗散的局部可解性。另一方面，我们在维纳空间中建立了足够大的粘性的全局适定性。这些结果是 Dias, Dyachenko \& Zakharov 系统（{\em Physics Letters A} 2008）对具有粘性的重力波建模的首次严格证明。