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1. Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case

   https://doi.org/10.1090/memo/1377  

   Bedrossian, Jacob ; Germain, Pierre ; Masmoudi, Nader ( September 2022 , Memoirs of the American Mathematical Society)

   This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re . In this work, we show that there is constant 0 > c 0 ≪ 1 0 > c_0 \ll 1 , independent of R e \mathbf {Re} , such that sufficiently regular disturbances of size ϵ ≲ R e − 2 / 3 − δ \epsilon \lesssim \mathbf {Re}^{-2/3-\delta } for any δ > 0 \delta > 0 exist at least until t = c 0 ϵ − 1 t = c_0\epsilon ^{-1} and in general evolve to be O ( c 0 ) O(c_0) due to the lift-up effect. Further, after times t ≳ R e 1 / 3 t \gtrsim \mathbf {Re}^{1/3} , the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ϵ − 1 t \approx \epsilon ^{-1} . Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the “lift-up effect ⇒ \Rightarrow streak growth ⇒ \Rightarrow streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature. 

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2. Vanishing viscosity limit for axisymmetric vortex rings

   https://doi.org/10.1007/s00222-024-01261-5  

   Gallay, Thierry ; Šverák, Vladimír ( July 2024 , Inventiones mathematicae)

   For the incompressible Navier-Stokes equations in R3 with low viscosity ν > 0, we consider the Cauchy problem with initial vorticity ω0 that represents an infinitely thin vortex filament of arbitrary given strength \Gamma supported on a circle. The vorticity field ω(x, t) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness√νt that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that ω(x, t) is well-approximated on a large time interval by ω_lin (x − a(t), t), where ω_lin(·, t) = exp(νt\Delta)ω0 is the solution of the heat equation with initial data ω0, and ˙a(t) is the instantaneous velocity given by Kelvin’s formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold’s geometric stability methods developed in the inviscid case ν = 0 to the slightly viscous situation. It turns out that although the geometric structures behind Arnold’s approach are no longer preserved by the equation for ν >0, the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold’s theory and interact favorably with the viscous terms. 

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3. On the Local Existence of Solutions to the compressible Navier–Stokes-Wave System with a Free Interface

   https://doi.org/10.1007/s00021-024-00861-8  

   Kukavica, Igor ; Li, Linfeng ; Tuffaha, Amjad ( March 2024 , Journal of Mathematical Fluid Mechanics)

   We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space$$H^{2+\epsilon }$$$H^{2+ϵ}$and the initial structure velocity is in$$H^{1.5+\epsilon }$$$H^{1.5+ϵ}$, where$$\epsilon \in (0,1/2)$$$ϵ\in (0,1/2)$.

    

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4. Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field

   https://doi.org/10.1007/s00220-023-04771-0  

   Costin, Ovidiu ; Costin, Rodica ; Jauslin, Ian ; Lebowitz, Joel L ( September 2023 , Communications in Mathematical Physics)

   Springer (Ed.)

   We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space x 0, the Schrödinger equation of the system is i∂t ψ = − 2 1 ∂x 2 ψ + (x)(U − E x cos ωt)ψ, t > 0, x ∈ R, where (x) is the Heaviside function and U > 0 is the effective confining potential (we choose units so that m = e = = 1). The amplitude E of the external electric field and the frequency ω are arbitrary. We prove existence and uniqueness of classical solutions of the Schrödinger equation for general initial conditions ψ(x, 0) = f (x), x ∈ R. When 2the initial condition is in L the evolution is unitary and the wave function goes to zero at any fixed x as t → ∞. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-L 2 initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all t > 0. For these initial conditions we show that the solution approaches in the large t limit a periodic state that satisfies an infinite set of equations formally derived, under the assumption that the solution is periodic by Faisal et al. (Phys Rev A 72:023412, 2005). Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior, as seen both analytically and numerically. It shows a steep increase in the current as the frequency passes a threshold value ω = ωc , with ωc depending on the strength of the electric field. For small E, ωc represents the threshold in the classical photoelectric effect, as described by Einstein’s theory. 

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5. Sample Complexity of Probabilistic Roadmaps via epsilon-nets

   Tsao, M ; Solovey, K ; Pavone, M ( September 2020 , IEEE International Conference on Robotics and Automation)

   We study fundamental theoretical aspects of probabilistic roadmaps (PRM) in the finite time (non-asymptotic) regime. In particular, we investigate how completeness and optimality guarantees of the approach are influenced by the underlying deterministic sampling distribution ${\X}$ and connection radius ${r>0}$. We develop the notion of ${(\delta,\epsilon)}$-completeness of the parameters ${\X, r}$, which indicates that for every motion-planning problem of clearance at least ${\delta>0}$, PRM using ${\X, r}$ returns a solution no longer than ${1+\epsilon}$ times the shortest ${\delta}$-clear path. Leveraging the concept of ${\epsilon}$-nets, we characterize in terms of lower and upper bounds the number of samples needed to guarantee ${(\delta,\epsilon)}$-completeness. This is in contrast with previous work which mostly considered the asymptotic regime in which the number of samples tends to infinity. In practice, we propose a sampling distribution inspired by ${\epsilon}$-nets that achieves nearly the same coverage as grids while using significantly fewer samples. 

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