This paper is concerned with the Navier-stokes equations with nonlinear perturbation in R^2,which studies the existence of solution,and gets the existence of the attractors.Finally,we discuss with limit-behavior of the Navier-stokes equation4 with nonlinear per-turbation,asα→0.
In this paper,we study the Minkowski valuations on the set of 1 or2 dimensional convex bodies compatible with a linear transformation and translations in some sense.We first introduce a kind of compatibility that all linear transformations(as Minkowski valuations)possess naturally.Then,we show that,under some natural conditions,monotone Minkowski valuations with such compatibility are exactly linear transformations.So we obtain a valuation characterization of linear transformations on Euclidean 1 or2-spaces.
In this paper,we investigate the translative containment measure for a convex domain K_i to contain,or to be contained in the homothetic copy of another convex domain tK_j(t≥0).Via the formulas of translative Blaschke and Poincare in integral formula,we obtain a Bonnesen-style symmetric mixed isohomothetic inequality.The Bonnesen-style symmetric mixed isohomothetic inequality obtained is known as Bonnesen-style inequality if one of the domains is a disc.As a direct consequence,we attain an inequality which strengthen the result proved by Bonnesen,Blaschke and Flanders.Furthermore,by the containment measure and Blaschke’s rolling theorem,we obtain the reverse Bonnesen-style symmetric mixed isohomothetic inequalities.These inequalities are the analogues of the known Bottema’s result in 1933.
