Turbulence is one of the most outstanding and still unanswered problems in
classical physics. The motion of viscous incompressible turbulent flows is
governed by the Navier-Stokes equations (NSEs). The mathematical study of
Weizmann scientists of this phenomenon and equations is strongly motivated
by physical assertions and experimental observations.
The question of whether the solutions of the three-dimensional NSEs breakdown in finite-time or remain nice indefinitely in time is one of the most challenging questions in applied analysis, since the pioneer work of J. Leray in the 1930s; and is one of the seven Millennium Problems of the Clay Mathematics Institute. Weizmann Institute scientist and collaborators proved in 1990 that three-dimensional solutions of the NSEs with helical symmetry remain nice indefinitely. Furthermore in 1994 they proved the long standing Onsager conjecture concerning the conservation of energy of non-classical solutions for the three-dimensional Euler equations of non-viscous fluids.
The NSEs form a basic building block in climate and planetary oceanic and
atmospheric dynamics models. Taking into account the thinness of the oceans
and atmosphere, in comparison the earth's diameter, Richardson introduced
in 1920s the "primitive equations" for climate and weather prediction.
Even though the reduced primitive equations are supposedly more accessible computationally they were thought, for long time, by many mathematical experts to be more challenging than the NSEs. It was not until 2005 when a Weizmann Institute scientist and collaborates succeeded to show that the solutions of three-dimensional primitive equations remain nice indefinitely. In 2012 they established another remarkable result showing that certain solutions of the non-viscous primitive equations breakdown in finite-time. Establishing similar results for the Navier-Stokes and Euler equations is the ultimate goal.