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We consider the motion of incompressible viscous fluids bounded above by afree surface and below by a solid surface in the $N$-dimensional Euclideanspace for $N\geq 2$ when the gravity is not taken into account. The aim of thispaper is to show the global solvability of the Naiver-Stokes equations with afree surface, describing the above-mentioned motion, in the maximal$L_p\text{-}L_q$ regularity class. Our approach is based on the maximal$L_p\text{-}L_q$ regularity with exponential stability for the linearizedequations, and solutions to the original nonlinear problem are alsoexponentially stable.

Global solvability of the Navier-Stokes equations with a free surface in the maximal $L_p\text{-}L_q$ regularity class