Approximate solutions are found computationally. Non-trivial closed-form solutions are hopeless. Existence of solutions is a little less hopeless. Existence

Approximate solutions are found computationally. Non-trivial closed-form solutions are hopeless. Existence of solutions is a little less hopeless. Existence of solutions over small time horizons has been proven. This is all in three space dimensions. There is a theorem stating that if global-time solutions exist and are unique, then computationally computed approximate solutions are 'good'. Moreover, they get better as you refine the computational domain.