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1. Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

   https://doi.org/10.1007/s00205-021-01651-6  

   Guo, Yan; Huang, Feimin; Wang, Yong (July 2021, Archive for Rational Mechanics and Analysis)

   Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic study of the viscous layer equations and the L2 to L∞ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations in half-space. 

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2. The inviscid limit for the 2D Navier-Stokes equations in bounded domains

   https://doi.org/10.3934/krm.2022004  

   Bardos, Claude W.; Nguyen, Trinh T.; Nguyen, Toan T.; Titi, Edriss S. (January 2022, Kinetic and Related Models)

   We prove the inviscid limit for the incompressible Navier-Stokes equations for data that are analytic only near the boundary in a general two-dimensional bounded domain. Our proof is direct, using the vorticity formulation with a nonlocal boundary condition, the explicit semigroup of the linear Stokes problem near the flatten boundary, and the standard wellposedness theory of Navier-Stokes equations in Sobolev spaces away from the boundary. 

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3. Recovery of the Nonlinearity From the Modified Scattering Map

   https://doi.org/10.1093/imrn/rnad243  

   Chen, Gong; Murphy, Jason (October 2023, International Mathematics Research Notices)

   Abstract We consider a class of one-dimensional nonlinear Schrödinger equations of the form $$ \begin{align*} & (i\partial_{t}+\Delta)u = [1+a]|u|^{2} u. \end{align*}$$For suitable localized functions $$a$$, such equations admit a small-data modified scattering theory, which incorporates the standard logarithmic phase correction. In this work, we prove that the small-data modified scattering behavior uniquely determines the inhomogeneity $$a$$. 

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4. Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

   Giordano Cotti, Alexander Varchenko (January 2021, Proceedings of symposia in pure mathematics)

   Novikov, Krichever (Ed.)

   In [TV19a] the equivariant quantum differential equation (qDE) for a projective space was considered and a compatible system of difference qKZ equations was introduced; the space of solutions to the joint system of the qDE and qKZ equations was identified with the space of the equivariant K-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant K-theory algebra. This paper is a continuation of [TV19a]. We describe the relation between solutions to the joint system of the qDE and qKZ equations and the topological-enumerative solution to the qDE only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant K-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant K-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani. 

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5. On the validity of quasilinear theory applied to the electron bump-on-tail instability

   https://doi.org/10.1063/5.0086442  

   Crews, D. W.; Shumlak, U. (April 2022, Physics of Plasmas)

   The accuracy of quasilinear theory applied to the electron bump-on-tail instability, a classic model problem, is explored with conservative high-order discontinuous Galerkin methods applied to both the quasilinear equations and to a direct simulation of the Vlasov–Poisson equations. The initial condition is chosen in the regime of beam parameters for which quasilinear theory should be applicable. Quasilinear diffusion is initially in good agreement with the direct simulation but later underestimates the turbulent momentum flux. The greater turbulent flux of the direct simulation leads to a correction from quasilinear evolution by quenching the instability in a finite time. Flux enhancement above quasilinear levels occurs as the phase space eddy turnover time in the largest amplitude wavepackets becomes comparable to the transit time of resonant phase fluid through wavepacket potentials. In this regime, eddies effectively turn over during wavepacket transit so that phase fluid predominantly disperses by eddy phase mixing rather than by randomly phased waves. The enhanced turbulent flux of resonant phase fluid leads, in turn, through energy conservation to an increase in non-resonant turbulent flux and, thus, to an enhanced heating of the main thermal body above quasilinear predictions. These findings shed light on the kinetic turbulence fluctuation spectrum and support the theory that collisionless momentum diffusion beyond the quasilinear approximation can be understood through the dynamics of phase space eddies (or clumps and granulations). 

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