#### Milestone Year

### 1990

#### On The Navier-Stokes and Euler Equations

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.