Open problems

980 registered problems.

OP-00567 Open

Matheron’s covariogram uniqueness problem in three dimensions

MG · AIM Open Applied Problems · convex bodies · covariograms · phase retrieval

OP-00568 Open

Broken-ray tomography with two smooth convex obstacles

AP · AIM Open Applied Problems · broken ray transforms · integral geometry · reflecting obstacles

OP-00569 Open

A data-independent time for global control of the defocusing cubic wave

OC · AIM Open Applied Problems · exact controllability · semilinear wave equations

OP-00570 Open

Exact optimal control in the scalar Witsenhausen benchmark

OC · AIM Open Applied Problems · team decision problems · Witsenhausen's counterexample

OP-00571 Solution claimed

Interior bounds for stable elliptic states with sign-changing reaction laws

AP · AIM Open Applied Problems · semilinear elliptic equations · stable solutions

OP-00572 Open

Fast dynamo action of the unit ABC flow

AP · ABC flows · AIM Open Applied Problems · fast dynamos

OP-00573 Open

Fourier’s law for a boundary-thermostated anharmonic chain

PR · AIM Open Applied Problems · anharmonic chains · heat conduction · nonequilibrium steady states

OP-00574 Open

Strict physicality for anisotropic singular Q-tensor minimizers

AP · AIM Open Applied Problems · Landau–de Gennes theory · Q-tensors · singular potentials

OP-00575 Open

Existence of an optimal asymmetric Bloch wall of prescribed angle

AP · AIM Open Applied Problems · domain walls · micromagnetics

OP-00576 Open

The optimal three-dimensional quadratic quantization constant

OC · AIM Open Applied Problems · centroidal Voronoi tessellations · lattice quantizers · optimal quantization

OP-00577 Open

Hard-sphere Boltzmann shock profiles at arbitrary Mach number

AP · AIM Open Applied Problems · Boltzmann equation · kinetic theory · shock profiles

OP-00578 Open

Bressan's logarithmic lower bound on the cost of mixing

AP · AIM Open Applied Problems · incompressible flows · mixing · transport equations

OP-00579 Open

Global classical continuation in the chemotaxis-consumption system

AP · AIM Open Applied Problems · chemotaxis-consumption systems · global classical solutions

OP-00580 Open

Finitely many radius sets for compact sphere packings

MG · AIM Open Applied Problems · compact packings · sphere packings

OP-00581 Open

Global attraction for complex-balanced reaction–diffusion with boundary equilibria

AP · AIM Open Applied Problems · chemical reaction networks · convergence to equilibrium · reaction–diffusion systems

OP-00582 Open

Global finite-energy compressible flow at the diatomic exponent

AP · AIM Open Applied Problems · compressible Navier–Stokes equations · weak solutions

OP-00583 Open

Higher regularity for concave stable nonlocal equations

AP · AIM Open Applied Problems · Bellman equations · fully nonlinear integro-differential equations

OP-00584 Open

A single blow-up point for singular extremal states on convex domains

AP · AIM Open Applied Problems · extremal solutions · singular sets

OP-00585 Open

One cubic NLS trajectory with unbounded Sobolev norm

AP · AIM Open Applied Problems · energy cascades · nonlinear Schrödinger equations

OP-00586 Open

Competitive exclusion by the slowest diffuser in a multispecies habitat

AP · AIM Open Applied Problems · competitive exclusion · evolution of dispersal · reaction-diffusion systems

OP-00587 Open

Do finite-energy elastic minimizers satisfy force balance?

AP · AIM Open Applied Problems · Euler–Lagrange equations · nonlinear elasticity · polyconvexity

OP-00588 Open

The attainable stiffness tensors of two-phase elastic composites

AP · AIM Open Applied Problems · composite materials · G-closures · homogenization

OP-00589 Open

Giga’s embedded curve-diffusion conjecture

AP · AIM Open Applied Problems · curve diffusion flow · embedded curves · geometric evolution equations

OP-00590 Open

Global regularity of the planar noncorotational FENE fluid

AP · AIM Open Applied Problems · FENE dumbbell model · Fokker–Planck equations · polymeric fluids

OP-00591 Open

Weak uniqueness for fourth-order aggregation–diffusion

AP · aggregation-diffusion equations · AIM Open Applied Problems · thin-film equations · Wasserstein gradient flows

OP-00592 Open

Hölder regularity for nondivergence kinetic jump equations

AP · AIM Open Applied Problems · Hölder estimates · integro-differential equations · kinetic equations

OP-00593 Open

Flatness of three-dimensional fractional minimizing cones for every interaction order

AP · AIM Open Applied Problems · fractional perimeters · minimal cones · nonlocal minimal surfaces

OP-00594 Open

Instantaneous spreading for fractional-pressure flow in several dimensions

AP · AIM Open Applied Problems · fractional pressure · infinite speed of propagation · porous medium equations

OP-00595 Open

Multidimensional weak uniqueness for porous-medium flow with fractional pressure

AP · AIM Open Applied Problems · fractional pressure · porous medium equations · uniqueness of weak solutions

OP-00596 Open

Bounded-Hessian Liouville rigidity in dimensions three and four

AP · AIM Open Applied Problems · fully nonlinear elliptic equations · Liouville theorems

OP-00597 Open

Uniqueness of Glimm–Lax solutions for arbitrarily rough bounded data

AP · AIM Open Applied Problems · entropy solutions · hyperbolic conservation laws

OP-00598 Open

Rigidity of positively curved shells above one-half Hölder regularity

DG · AIM Open Applied Problems · convex integration · convex surfaces · isometric embeddings

OP-00599 Open

Global regularity of two-dimensional ideal incompressible MHD

AP · AIM Open Applied Problems · global regularity · incompressible flows · magnetohydrodynamics

OP-00600 Open

The exact minimum ropelength of a trefoil

GT · AIM Open Applied Problems · ideal knots · ropelength

OP-00601 Open

Uniqueness of the homogeneous cooling state for inelastic hard spheres

AP · AIM Open Applied Problems · granular gases · inelastic Boltzmann equations · self-similar solutions

OP-00602 Open

Large-data global smoothness for the spatially inhomogeneous Boltzmann equation

AP · AIM Open Applied Problems · Boltzmann equation · classical solutions · kinetic equations

OP-00603 Open

Smooth unforced inviscid Boussinesq evolution on the whole plane

AP · AIM Open Applied Problems · Boussinesq equations · finite-time blowup · global regularity

OP-00604 Open

Quantization of radial finite-time collapse in fully parabolic Keller–Segel

AP · AIM Open Applied Problems · chemotactic collapse · finite-time blow-up · Keller–Segel system

OP-00605 Open

The three-dimensional lattice-periodic Kelvin problem

AP · AIM Open Applied Problems · Kelvin problem · lattice tilings · minimal partitions

OP-00606 Open

A length-independent mean energy bound for Kuramoto–Sivashinsky

AP · AIM Open Applied Problems · energy bounds · Kuramoto–Sivashinsky equation · spatiotemporal chaos

OP-00607 Open

Are bounded ancient Landau–Coulomb distributions necessarily Maxwellians?

AP · AIM Open Applied Problems · ancient solutions · Landau equation · Liouville theorems

OP-00608 Open

Landau–Coulomb smoothing controlled only by macroscopic quantities

AP · AIM Open Applied Problems · Landau equation · regularization estimates

OP-00609 Open

The bulk ground-state energy of the planar Lennard–Jones system

MP · AIM Open Applied Problems · crystallization · lattice energies · Lennard–Jones potentials

OP-00610 Open

The three-dimensional steady Leray problem with unrestricted boundary fluxes

AP · AIM Open Applied Problems · Leray problem · Navier–Stokes equations · steady flows

OP-00611 Open

Global weak flow of three-dimensional nematic directors without a hemisphere restriction

AP · AIM Open Applied Problems · Ericksen–Leslie system · nematic liquid crystals · weak solutions

OP-00612 Open

Instantaneous strict separation for three-dimensional local Cahn–Hilliard

AP · AIM Open Applied Problems · Cahn–Hilliard equations · logarithmic potentials · strict separation

OP-00613 Open

Uniqueness of all weak dual solutions of fractional porous-medium flow

AP · AIM Open Applied Problems · fractional porous medium equations · Riemannian manifolds · weak dual solutions

OP-00614 Open

Global classical Maxwell–Stefan diffusion with arbitrary pair friction

AP · AIM Open Applied Problems · cross-diffusion systems · gas mixtures · Maxwell–Stefan equations

OP-00615 Open

Nonlinear Landau damping near a Maxwellian on the whole space

AP · AIM Open Applied Problems · Landau damping · Vlasov–Poisson equations

OP-00616 Open

Regular MEMS extremal states without an asymptotic curvature condition

AP · AIM Open Applied Problems · extremal solutions · singular nonlinearities