Authors: Ángel Castro (ICMAT-CSIC), Diego Córdoba (ICMAT-CSIC) and Javier Gómez-Serrano
Source: American Mathematical Society
Date of publication: July 2020
The Surface Quasi-Geostrophic (SQG) equation was originally derived for modelling the movement of large masses of air in the atmosphere and, in particular, for frontogenesis, the formation of sharp fronts between masses of hot and cold air. This is an equation that reduces the dynamics of fluids from three to two dimensions, by eliminating altitude and working on a plane –in this case, the surface of the Earth, taken over a small region so that its curvature can be disregarded. The equation is given in terms of temperature and is an evolution equation. Given some initial conditions of temperature for an initial time, the solution of the equation yields as a result the temperature over subsequent time. Once the temperature is known, the movement of the fluid can be determined. In this paper the study is conducted on all the plane ², although studies exist in which bounded regions are also considered.
Despite the use that the SQG equation may have in other sciences, this is not the characteristic that has prompted the interest of the authors of this present study. Peter Constantin, Andrew Majda Esteban Tabak pointed out that the structure of the equation is similar to that of the Euler equation for an incompressible fluid. Indeed, in both the Euler 3D equation for the formulation of vorticity and the equation for the gradient of the temperature on the SQG equation, a transport term appears, with a velocity given by an operator of degree -1 in derivatives and also a quadratic term that involves singular integrals. However, the Euler equation considers the evolution of a vector of three components in 3 dimensions and an SQG the evolution of a scalar in two dimensions, the latter therefore being easier to manage. In turn, the Euler equation is a limit case of the Navier-Stokes equation, in which the viscosity of the fluid is taken to be zero. The burning issue in fluid mechanics is to determine whether the Navier-Stokes solutions are globally regular or if on the other hand they develop singularities in finite time. This question remains unresolved in both the Navier-Stokes equations and the Euler equation. Constantin, Majda and Tabak studied the SQG equation with the aim of gaining a greater understanding of the Euler equation, and the authors of this paper have followed in their footsteps. The search for singularities in the SQG equation could provide a better understanding of the mechanisms governing the behaviour of the Euler equation.
Until the publication of the paper, it was known that the SQG equation remained smooth for a short time interval, regardless of the initial conditions, but the only known examples in which the solutions remained smooth over all time were the trivial stationary solutions, when the temperature is given by any radial function.
In this paper, the authors show that families of initial conditions exist such that the solution is smooth over all time, without the development of any singularity. Such solutions consist of a global rotation with constant angular velocity of the temperature. To achieve the proof, the problem is reduced analytically to check an open condition that is then rigorously demonstrated by means of computer-assisted proofs using interval arithmetic.
The analytical part is based on the application of the Crandall-Rabinowitz theorem, which involves the study of the spectrum of the operator given by the linearization of the SQG equation. It is this study that requires the obtention of certain bounds of the eigenvalues of the linear operator, which, module small errors that can be estimated with pen and paper, are given by enormous but explicit expressions, and it is the obtention of these bounds for which the computer is employed. The philosophy of interval arithmetic is as follows: a computer is unable to yield rigorously the result of any operation because only a finite amount of numbers is available in its register. Nevertheless, this register of numbers can be used to work with a finite number of intervals covering a large part of the real number line, and it is with these intervals that rigorous estimates of complex operations can be obtained. For example, if we wish to add A plus B, it may be said that A is in interval I and B is in interval J; the sum of I plus J produces the interval K, and then it may be rigorously stated that the result of A plus B is within K. If the interval K is sufficiently small, whatever this may mean in your problem, you have won.