Mathematics Problem Archive

Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?...

Do solutions to the 3D Euler equations for incompressible fluid flow remain smooth for all time, given smooth initial data?...

Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals....

Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time?...

Is the Mandelbrot set locally connected?...

Does every regular compact contact-type level set carry a periodic orbit?...

If a billiard table is strictly convex and integrable, must its boundary be an ellipse?...

Do smooth solutions to Navier-Stokes equations exist globally in 3D? Or do finite-time singularities occur?...

Find useful ways of verifying polyconvexity and quasiconvexity for stored-energy functions arising in anisotropic nonlinear elasticity....

Determine when the minimizer $y^*$ in Theorem 2.1 of the source is smooth....

Prove or disprove that, under reasonable growth conditions on $W$, an energy-minimizing deformation satisfies $\det Dy^*(x)\ge\varepsilon>0$....

Devise general methods for proving the existence of local but nonglobal minimizers and other weak equilibria in nonlinear elastostatics....

Develop a qualitative dynamics for dynamic theories of elasticity....

Establish the status of elasticity theory with respect to atomistic models....

For free-energy functions $\psi(A,\theta)$ of elastic crystals, determine boundary conditions under which the minimum is attained and conditions under...

Is the global attractor of the periodically forced two-dimensional Navier–Stokes equations conjugate to a smooth finite-dimensional dynamical system? ...