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1. Three-dimensional General-relativistic Simulations of Neutrino-driven Winds from Rotating Proto-neutron Stars

   https://doi.org/10.3847/1538-4357/ac69da  

   Desai, Dhruv ; Siegel, Daniel M. ; Metzger, Brian D. ( May 2022 , The Astrophysical Journal)

   Abstract We explore the effects of rapid rotation on the properties of neutrino-heated winds from proto-neutron stars (PNS) formed in core-collapse supernovae or neutron-star mergers by means of three-dimensional general-relativistic hydrodynamical simulations with M0 neutrino transport. We focus on conditions characteristic of a few seconds into the PNS cooling evolution when the neutrino luminosities obey L ν e + L ν ¯ e ≈ 7 × 10 51 erg s −1 , and over which most of the wind mass loss will occur. After an initial transient phase, all of our models reach approximately steady-state outflow solutions with positive energies and sonic surfaces captured on the computational grid. Our nonrotating and slower rotating models (angular velocity relative to Keplerian Ω/Ω K ≲ 0.4; spin period P ≳ 2 ms) generate approximately spherically symmetric outflows with properties in good agreement with previous PNS wind studies. By contrast, our most rapidly spinning PNS solutions (Ω/Ω K ≳ 0.75; P ≈ 1 ms) generate outflows focused in the rotational equatorial plane with much higher mass-loss rates (by over an order of magnitude), lower velocities, lower entropy, and lower asymptotic electron fractions, than otherwise similar nonrotating wind solutions. Although such rapidly spinning PNS are likely rare in nature, their atypical nucleosynthetic composition and outsized mass yields could render them important contributors of light neutron-rich nuclei compared to more common slowly rotating PNS birth. Our calculations pave the way to including the combined effects of rotation and a dynamically important large-scale magnetic field on the wind properties within a three-dimensional GRMHD framework. 

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2. Probability flow solution of the Fokker–Planck equation

   https://doi.org/10.1088/2632-2153/ace2aa  

   Boffi, Nicholas M. ; Vanden-Eijnden, Eric ( July 2023 , Machine Learning: Science and Technology)

   The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its ‘score’), and so isa-prioriunknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals.

    

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3. Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity

   Pauthier, Antoine ; Poláčik, Peter ( August 2022 , Partial Differential Equations and Applications)

   Abstract This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation $$u_t=u_{xx}+f(u)$$ u t = u xx + f ( u ) on the real line whose initial data $$u_0=u(\cdot ,0)$$ u 0 = u ( · , 0 ) have finite limits $$\theta ^\pm $$ θ ± as $$x\rightarrow \pm \infty $$ x → ± ∞ . We assume that f is a locally Lipschitz function on $$\mathbb {R}$$ R satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u ( x ,  t ) as $$t\rightarrow \infty $$ t → ∞ . In the first two parts of this series we mainly considered the cases where either $$\theta ^-\ne \theta ^+$$ θ - ≠ θ + ; or $$\theta ^\pm =\theta _0$$ θ ± = θ 0 and $$f(\theta _0)\ne 0$$ f ( θ 0 ) ≠ 0 ; or else $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is a stable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of $$u(\cdot ,t)$$ u ( · , t ) as $$t\rightarrow \infty $$ t → ∞ are steady states. The limit profiles, or accumulation points, are taken in $$L^\infty _{loc}(\mathbb {R})$$ L loc ∞ ( R ) . In the present paper, we take on the case that $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is an unstable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on $$u(\cdot ,t)$$ u ( · , t ) is that it is nonoscillatory (has only finitely many critical points) at some $$t\ge 0$$ t ≥ 0 . Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal. 

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4. 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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5. An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals

   https://doi.org/10.1093/imanum/draa102  

   Cockburn, Bernardo ; Xia, Shiqiang ( January 2021 , IMA Journal of Numerical Analysis)

   Abstract We present the first a priori error analysis of a new method proposed in Cockburn & Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $u$ is the solution of a steady-state diffusion problem, the standard approximation $J(u_h)$ converges with order $h^{2k+1}$, where $u_h$ is the Hybridizable Discontinuous Galerkin approximation to $u$ with polynomials of degree $k>0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $h^{4k}$, and can be computed by only using twice the computational effort needed to compute $J(u_h)$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $u$ or the functional $J(\cdot )$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods. 

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