Polynomial sum of squares in fluid dynamics: a review with a look ahead

Polynomial sum of squares in fluid dynamics: a review with a look ahead.

The first part of this paper reviews the application of the sum-of-squares-of-polynomials technique to the problem of global stability of fluid flows. It describes the known approaches and the latest results, in particular, obtaining for a version of the rotating Couette flow a better stability range than the range given by the classic energy stability method. The second part of this paper desc...

Sum of Squares and Polynomial Convexity

The notion of sos-convexity has recently been proposed as a tractable sufficient condition for convexity of polynomials based on sum of squares decomposition. A multivariate polynomial p(x) = p(x1, . . . , xn) is said to be sos-convex if its Hessian H(x) can be factored as H(x) = M (x) M (x) with a possibly nonsquare polynomial matrix M(x). It turns out that one can reduce the problem of decidi...

Conditions for a Real Polynomial to Be Sum of Squares

We provide explicit conditions for a polynomial f of degree 2d to be a sum of squares (s.o.s.), stated only in terms of the coefficients of f , i.e. with no lifting. All conditions are simple and provide an explicit description of a convex polyhedral subcone of the cone of s.o.s. polynomials of degree at most 2d. We also provide a simple condition to ensure that f is s.o.s., possibly modulo a c...

Sum of Squares Programs and Polynomial Inequalities

How can one find real solutions (x1, x2)? How to prove that they do not exist? And if the solution set is nonempty, how to optimize a polynomial function over this set? Until a few years ago, the default answer to these and similar questions would have been that the possi­ ble nonconvexity of the feasible set and/or objective function precludes any kind of analytic global results. Even today, t...

A look back, a look ahead.