gn.general topology - Applications of Brouwer's fixed point theorem

In many nonlinear equations, the existence of a solution (but not its uniqueness) follows from a topological argument in the vein of BFP Theorem. The BFP is at work especially when the equation is posed in some finite dimensional vector space, and you can establish an a priori estimate of the size of the solution. This means that you know a ball $B_R$ containing all the solutions.

The most important example of this situation is the stationary Navier-Stokes equation, with Dirichlet condition $u=0$ on the boundary of the domain. Of course, the ambient space is infinite dimensional, so you first establish the existence of an approximate solution in a subspace of dimension $n$ (Galerkin procedure); this is where you use the BFP Theorem, or its equivalent form that a continuous vector field over $B_R$ which is outgoing on $partial B_R$ must vanish somewhere. Then passing to the limit as $nrightarrowinfty$ is pedestrian.

The BFP Theorem is a consequence of the fact that the Euler-Poincaré characteristic of the ball is non-zero. There are counterparts when you work on a compact manifold (with boundary) whose EPC is non-zero. This happened to me in a very interesting way. I considered the free fall of a rigid body in water filling the entire space. The mathematical problem is a coupling between Navier-Stokes and the Euler equation for the top. I looked at a permanent regime, in which the solid body has a time-independent velocity field, and that of the fluid is time-independent as well, once you consider it in the moving frame attached to the solid. The difficulty is that you don't know a priori the direction of the vertical axis (the direction of gravity) in this frame. After a Galerkin procedure, the problem reduces to the search of a zero of a tangent vector field over $B_Rtimes S^2$. This vector field is outgoing on the boundary $partial B_Rtimes S^2$. Because
$$EP(B_Rtimes S^2)=EP(B_R)cdot EP(S^2)=1cdot2ne0,$$
such a zero exists. Therefore the permanent regime does exist. Remark that because $EP=2$, we even expect an even number of solutions when counting multiplicities, at least at each level of the Galerkin approximation.