J. math. fluid mech. 1 (1999) 235–281 1422-6928/99/030235-47 $ 1.50+0.20/0 c  1999 Birkh¨auser Verlag, Basel Journal of Mathematical Fluid Mechanics Interior Regularity for Solutions to the Modified Navier–Stokes Equations G. A. Seregin Communicated by O. L. Ladyzhenskaya Abstract. We discuss interior regularity of solutions to the three-dimensional modified Navier– Stokes equations. In particular, we formulatesufficient conditions that guarantee the local H¨older continuity of the velocity gradient. Mathematics Subject Classification (1991). 35K, 76D. Keywords. The modified Navier–Stokes equations, initial-boundary value problems, interior regularity, Hausdorff’s dimension. 1. Introduction The crucial question in the mathematical theory of viscous incompressible fluids is as follows: what are the conditions that guarantee the global unique solvability of the given initial-boundary value problems? For the three-dimensional Navier– Stokes equations, the existence of weak solutions was established long time ago (for details see [5]). However, it is still an open problem whether a weak solution is unique or not. It is only known (see [5]) that if the set of weak solutions contains a smooth one, then we have the uniqueness. To the best of my knowledge the most deep results on regularity of weak soltuions were obtained in the paper [1]. The authors of the above paper proved the existence of a so-called suitable weak solution. The latter is regular on an open set of full measure, whose Hausdorff’s dimension is not greater than 1. Unfortunately, this is not enough to prove the uniqueness of weak solutions. Moreover, some of the specialists in the mathemat- ical theory of the Navier–Stokes equations (see, for example, [5]) do not believe that such uniqueness takes place. In [2]–[4] O. A. Ladyzhenskaya introduced the so-called modified Navier–Stokes