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1. Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions

   https://doi.org/10.58997/ejde.2025.43  

   Bandyopadhyay, Shalmali; Lewis, Thomas; Mavinga, Nsoki (January 2025, Electronic Journal of Differential Equations)

   We establish the existence of maximal and minimal weak solutions   between ordered pairs of weak sub- and super-solutions for a coupled  system of elliptic equations with quasimonotone nonlinearities on the  boundary. We also formulate a finite difference method to approximate the  solutions and establish the existence of maximal and minimal approximations  between ordered pairs of discrete sub- and super-solutions.   Monotone iterations are formulated for constructing the maximal and minimal  solutions when the nonlinearity is monotone.  Numerical simulations are used to explore existence, nonexistence,  uniqueness and non-uniqueness properties of positive solutions.  When the nonlinearities do not satisfy the monotonicity condition, we prove the existence of weak maximal and minimal solutions using Zorn’s  lemma and a version of Kato’s inequality up to the boundary.  For more information see https://ejde.math.txstate.edu/Volumes/2025/43/abstr.html 

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2. Rough hypoellipticity for the heat equation in Dirichlet spaces

   https://doi.org/10.1002/mana.202100014  

   Hou, Qi; Saloff‐Coste, Laurent (January 2023, Mathematische Nachrichten)

   Abstract This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms, which satisfy mild assumptions concerning (1) the existence of cut‐off functions, (2) a local ultracontractivity hypothesis, and (3) a weak off‐diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussions on the existence of locally bounded heat kernel; structure results for ancient (local weak) solutions of the heat equation. 

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3. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

   https://doi.org/10.3934/krm.2022003  

   Dong, Hongjie; Guo, Yan; Yastrzhembskiy, Timur (January 2022, Kinetic and Related Models)

   We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the \begin{document}$$ S_p $$\end{document} estimate of [7], we prove regularity in the kinetic Sobolev spaces \begin{document}$$ S_p $$\end{document} and anisotropic Hölder spaces for such weak solutions. Such \begin{document}$$ S_p $$\end{document} regularity leads to the uniqueness of weak solutions. 

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4. Uniqueness and Weak-BV Stability for $$2\times 2$$ Conservation Laws

   https://doi.org/10.1007/s00205-022-01813-0  

   Chen, Geng; Krupa, Sam G.; Vasseur, Alexis F. (September 2022, Archive for Rational Mechanics and Analysis)

   Abstract Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of smallBVfunctions which are global solutions of this equation. For any smallBVinitial data, such global solutions are known to exist. Moreover, they are known to be unique amongBVsolutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not evenBV) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-BVuniqueness result, where only one of the solutions is supposed to be smallBV, and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in theBVtheory, in the case of systems with 2 unknowns. The method is$$L^2$$ $L^2$based, and builds up from the theory of a-contraction with shifts, where suitable weight functionsaare generated via the front tracking method. 

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5. Weak solutions for a poro-elastic plate system

   https://doi.org/10.1080/00036811.2021.1953483  

   Gurvich, Elena; Webster, Justin T. (July 2021, Applicable Analysis)

   null (Ed.)

   We consider a recent plate model obtained as a scaled limit of the three-dimensional Biot system of poro-elasticity. The result is a ‘2.5’-dimensional linear system that couples traditional Euler–Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We allow the permeability function to be time dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so-called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions for implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability function. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate and exposits a variety of interesting related models and associated analytical investigations. 

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