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1. Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

   https://doi.org/10.3934/mine.2023034  

   Silvestre, Luis; Snelson, Stanley (January 2022, Mathematics in Engineering)

   The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global smooth solutions for the non cutoff Boltzmann equation when the initial data is close to equilibrium. We include a short-time existence result for polynomially-weighted $$ L^\infty $$ initial data. From this, we deduce that if the initial data is sufficiently close to a Maxwellian in this norm, then a smooth solution exists globally in time. 

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2. On optimality and bounds for internal solutions generated from boundary data-driven Gramians.

   https://doi.org/10.1088/1361-6420/ae0fbe  

   Druskin, Vladimir; Moskow, Shari; Zaslavskiy, Mikhail (October 2025, Inverse Problems)

   Abstract We consider the computation of internal solutions for a time domain plasma wave equation with unknown coefficients from the data obtained by sampling its transfer function at the boundary. The computation is performed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian. We show that this approximation is asymptotically close to the projection of the true internal solution onto the subspace of background snapshots. This allows us to derive a generally applicable bound for the error in the approximation of internal fields from boundary data for a time domain plasma wave equation with an unknown potential $$q$$. For general $$q\in L^\infty$$, we prove convergence of these data generated internal fields in one dimension for two examples of initial waves. The first is for piecewise constant initial data and sampling $$\tau$$ equal to the pulse width. The second is piecewise linear initial data and sampling at half the pulse width. We show that in both cases the data generated solutions converge in $L^2$ at order $$\sqrt{\tau}$$. We present numerical experiments validating the result and the sharpness of this convergence rate. 

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3. Instability of the soliton for the focusing, mass-critical generalized KdV equation

   https://doi.org/10.3934/dcds.2021171  

   Dodson, Benjamin; Gavrus, Cristian (January 2022, Discrete & Continuous Dynamical Systems)

   In this paper we prove instability of the soliton for the focusing, mass-critical generalized KdV equation. We prove that the solution to the generalized KdV equation for any initial data with mass smaller than the mass of the soliton and close to the soliton in \begin{document}$$ L^{2} $$\end{document} norm must eventually move away from the soliton. 

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4. Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation

   https://doi.org/10.1353/ajm.2023.a913295  

   Farah, Luiz Gustavo; Holmer, Justin; Roudenko, Svetlana; Yang, Kai (December 2023, American Journal of Mathematics)

   Abstract: We consider the quadratic Zakharov-Kuznetsov equation $$\partial_t u + \partial_x \Delta u + \partial_x u^2=0$$ on $$\Bbb{R}^3$$. A solitary wave solution is given by $Q(x-t,y,z)$, where $$Q$$ is the ground state solution to $$-Q+\Delta Q+Q^2=0$$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $$Q$$ in the energy space, evolves to a solution that, as $$t\to\infty$$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $$x>\delta t-\tan\theta\sqrt{y^2+z^2}$$ for $$0\leq\theta\leq{\pi\over 3}-\delta$$. 

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5. Fully discrete pointwise smoothing error estimates for measure valued initial data

   https://doi.org/10.1051/m2an/2023076  

   Leykekhman, Dmitriy; Vexler, Boris; Wagner, Jakob (September 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interiorL^∞error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interiorL^∞error estimates forL²initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements. 

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