Approximate weak solution

A classical (or strong) solution to a differential equation is a function satisfying the differential equation pointwise. For some equations no proof of existence of classical solutions is known, for example this is the case for the Navier-Stokes equations describing viscous fluid flow.

The open problem of proving the existence of classical solutions to Navier-Stokes equations is listed as one of the Clay $1 million Prize problems, and to similarly proving existence of classical solutions to the inviscid Euler equations is regarded even harder by the mathematics community.

Exact and approximate weak solution

On the other hand, the mathematician Jean Leray in 1934 [1] proved the existence of so called weak solutions to the Navier-Stokes equations, satisfying the equations in mean value, not pointwise.

In [2] it is shown that for incompressible flow it is possible to use adaptive finite element methods (FEM) to compute approximate weak solutions to both the Navier-Stokes equations and the Euler equations. The character of an approximate weak solution is different from a classical solution. For example, a weak solution to the Euler equations can develop vorticity and turbulence, contrary to a classical solution where one can prove that this is impossible using the pointwise satisfaction of the equations by a classical solution.

The approximate weak solutions to the Euler equations in [2], dissipating kinetic energy through turbulence, are related to other types of weak solutions appearing in the mathematics literature, such as so called dissipative weak solutions [3].

References

[1] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace., Acta Mathematica, 63 (1934), pp. 193-248.

[2] J.Hoffman and C.Johnson, Computational Turbulent Incompressible Flow, Springer, 2007.

[3] J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible euler and Navier-Stokes solutions, Nonlinearity, (2000), pp. 249-255.