38 problems

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain equipped with the conformal metric $g=\omega\lvert dz\rvert^2$, where…

Let $L>0$, $M\in\mathbb{R}\setminus\{0\}$, and $\varepsilon>0$. Consider the transport–diffusion equation $u_t+M u_x=\varepsilon u_{xx}$ on $(0,T)\times(0,L)$, with a Dirichlet bou…

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let $\alpha,\beta$ be Kähler classes on $X$. Set $\mu=\frac{n\,\alpha^{n-1}\cdot\beta}{\alpha^n}$. The Datar–Mete…

For $n\ge 2$, fix an incident direction $\theta_0\in\mathbb S^{n-1}$. Determine whether a compactly supported potential $q$ is uniquely determined by its far-field pattern…

Let $n\geq 2$, let $u$ solve $\partial_t^2u-\Delta_xu=0$ on $\mathbb{R}^{n+1}$ with $u(\cdot,0)=u_0\in H^s(\mathbb{R}^n)$ and $\partial_tu(\cdot,0)=u_1\in H^{s-1}(\mathbb{R}^n)$, w…

Conjecture: Every asymptotically flat, stationary, axisymmetric solution $(\mathcal{M},g,F)$ of the Einstein--Maxwell equations representing a regular electrovacuum black hole is u…

For every $n\ge 3$, let $0<r<1$ and let $h:\mathbb{A}(r,1)\to\mathbb{A}(R,1)\subset\mathbb{R}^n$ be an onto homeomorphism whose coordinate functions are harmonic, where…

Let $N\ge 3$, let $\mathbb{R}^N_+=\{x\in\mathbb{R}^N:x_N>0\}$, and let $0\in\partial\mathbb{R}^N_+$. Consider positive solutions of the critical problem … with an isolated singular…

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.