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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. 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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3. 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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4. 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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5. Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation

   https://doi.org/10.1142/S0218202520500396  

   Rodríguez, Nancy; Winkler, Michael (October 2020, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Ref. 29 [M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267]. The focus here is on the question of how far a certain porous medium enhancement in the random diffusion of criminal agents may exert visible relaxation effects. It is shown that sufficient regularity of the non-negative source terms in the system and a sufficiently strong nonlinear enhancement ensure that a corresponding Neumann-type initial–boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is globally bounded under mild additional conditions on the source terms. These results are supplemented by numerical evidence which illustrates smoothing effects in solutions with sharply structured initial data in the presence of such porous medium-type diffusion and support the existence of singular structures in the linear diffusion case, which is the type of diffusion proposed in Ref. 29. 

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