Questions I’ll Never Answer

This section of my blog exists to compile the ever-growing list of mathematical & scientific questions that I heavily suspect I’ll never be able to answer. Therefore, if you think you know the answer to these (or think you know someone who does), please let me know!

  • Does this Strichartz estimate hold for nonlinear partial differential equations if they asymptotically approximate standard wave equations as t \lor |\mathbf{x}| \rightarrow \infty?
  • Is there a principle in non-equilibrium statistical mechanics that restricts or prohibits a nonlinear dependence of stress on strain rate for an isotropic, inelastic fluid?
  • How does the existence of compressible phenomena in fluids rigorously challenge or complement the notion that viscous effects represent a “diffusion of momentum density”?
  • Is there a useful extension of operator theorems (Fredholm alternative, etc.) for systems where the objects these operators would act on do not form a vector space, a.k.a. probability “vectors”?
  • Can a broad control-theoretic framework be developed for manipulating the dynamics of systems through state space as modeled by Boltzmann-like equations?
  • How does the Langevin equation emerge from the temporal scale separation of the viscous and noise-like processes, and what do the stochastic dynamics of a Brownian particle look like when these scales are only approximately separated?
  • What are the minimal extra assumptions required to derive classical electrodynamics from some number of Reynolds transport theorems over a Minkowski spacetime?
  • Are there simple analogues of Norton’s dome-like potentials that trigger non-uniqueness in low-Reynolds number fluid dynamics?
  • Can the framework for Hamiltonian dynamics be altered to model dynamical ensembles with nonholonomic constraints, and can this be used to generate a theory of quantum/relativistic dynamics for nonholonomic dynamical ensembles?
  • Is there a useful analogue for Wiener’s tauberian theorem that considers functions over finite domains?
  • Is there a “clean” proof for the strange behavior of Borwein integrals using only arguments from complex analysis?
  • Is there a way to derive continuum mechanics from non-equilibrium statistical mechanics using temporal scale-separation arguments, and would these arguments hold or fail in turbulent regimes?
  • What class of linear differential operators admits mean-value theorems over generalized balls (and how can you systematically produce them)?
  • Can the sequence T_{1} = a, T_{i+1} = F_{T_{i}} contain only primes for some a \in \mathbb{P} \setminus \{5\} ?
  • Can the function representing solutions to y' = sin(xy) for large y(0) be described using elementary functions?
  • Is there a formal connection between projection operators and quantitative logical deduction systems, and if so, what is it?
  • Can a Cauchy sequence of convolution approximations be developed generally without a need for a connection to random variables?
  • Can all local transport phenomena be fully described using analogues of convection & diffusion?