31 problems

For incompressible Navier-Stokes on $\mathbb{T}^d$, $d=2,3$, is the system approximately controllable and/or controllable in finite-dimensional projections by a localized degenerat…

Prove existence of a regular reflection for compressible flow against a wedge without assuming that the flow is irrotational.

Does the compressible Euler system in odd spatial dimension admit a nontrivial smooth eternal solution having finite nonzero mass and energy?

For compressible Navier–Stokes equations with constant viscosities near vacuum, does the unphysical one-dimensional consequence identified by Hoff and Serre have a multidimensional…

Develop a compensated-compactness calculus for symmetric matrices when compensated compactness yields only inequalities. As a first application, prove complete continuity of the se…

Develop a global-in-time theory of the Cauchy problem for the one-dimensional Euler–Fourier system for initial data constrained only by finite energy and entropy, possibly with loc…

For viscous incompressible flow in $\mathbb{R}^3\setminus B$ past a fixed obstacle $B$, with velocity approaching a nonzero constant vector at infinity, obtain realistic rigorous u…

Is the global attractor of the periodically forced two-dimensional Navier–Stokes equations conjugate to a smooth finite-dimensional dynamical system? Can its transient dynamics be…

Establish global regularity, or exhibit breakdown, for solutions of the Vlasov–Maxwell equations from appropriate smooth initial data.

Give a rigorous derivation of models of rods, plates, and shells from three-dimensional elasticity as thickness tends to zero.

For free-energy functions $\psi(A,\theta)$ of elastic crystals, determine boundary conditions under which the minimum is attained and conditions under which it is not.

For the set of energy-minimizing gradients $K(\theta)$ defined in the source, determine its quasiconvex hull $K(\theta)^{qc}$ for $\theta\le\theta_c$.

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

Develop criteria for dynamic stability and instability of equilibria in nonlinear elasticity.

Develop a qualitative dynamics for dynamic theories of elasticity.

Prove global existence and uniqueness for suitable initial-boundary-value problems in dynamic nonlinear elasticity.

Clarify the status of models based on the fracture energy functional (2.31) in the source relative to classical fracture and nonlinear elastostatics.

Develop local and global bifurcation theories for nonlinear elastostatics with mixed displacement-traction boundary conditions.

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

Prove or disprove uniqueness of sufficiently smooth equilibrium solutions for pure-displacement problems in homogeneous bodies homeomorphic to a ball when $W$ is strictly polyconve…

Justify the Ciarlet-Nečas minimization problem, or an appropriate modification, in situations involving smooth self-contact.

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

Prove or disprove that, under reasonable growth conditions on $W$, energy minimizers satisfy the weak Euler-Lagrange equations.

Can the Lavrentiev phenomenon occur for elastostatics under growth conditions ensuring that all finite-energy deformations are continuous?

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