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We consider the stationary Navier-Stokes equations in the n-dimensional torus T n for n 3. We show the existence and uniqueness of solutions in homogeneous toroidal Besov spacesWe can show its well-posedness by a similar method to that of Kaneko-Kozono-Shimizu (Indiana Univ. Math. J.), which has investigated the same problem in homogeneous Besov spaces on R n . Our advantage is to prove the ill-posedness in the critical exponents like p = n, 2 < q ∞ and n < p ∞, 1 q ∞. Indeed in such cases of p and q, there exists a sequence of external forces which converges to zero in Ḃ−3+ n p p,q (T n ) and yields a sequence of solutions which does not converge to zero in Ḃ−1 ∞,∞ (T n ). We can show this ill-posedness by constructing the sequence of external forces, as similar to those of initial data proposed by Yoneda (2010 J. Funct. Anal. 258 3376-87) in the non-stationary problem.

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Brand switching data among 12 margarine brands were analyzed by the asymmetric multidimensional scaling based on the singular value decomposition. A two-dimensional result was adopted as the solution. A configuration based on the left and right singular vectors is given along each dimension. The left singular vector represents an outward tendency of switching from the corresponding brand to the other brands, and the right singular vector represents an inward tendency of being switched to the corresponding brand from the other brands. The configuration along Dimension 1 shows that the three brands with the larger market share compete vigorously with each other. The configuration along Dimension 2 classifies 12 brands into two groups; the brand switching between two groups is small, while that within each group is large.

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