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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. Global Existence of Weak Solutions for Compresssible Navier—Stokes—Fourier Equations with the Truncated Virial Pressure Law

   https://doi.org/10.2478/caim-2023-0002  

   Bresch, Didier; Jabin, Pierre—Emmanuel; Wang, Fei (January 2023, Communications in Applied and Industrial Mathematics)

   Abstract This paper concerns the existence of global weak solutionsá la Lerayfor compressible Navier–Stokes–Fourier systems with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given. 

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3. Dynamics of a state-dependent delay-differential equation

   https://doi.org/10.14232/ejqtde.2025.1.37  

   Gedeon, Tomas; Humphries, Antony; Mackey, Michael; Walther, Hans-Otto; Wang, Zhao (January 2025, Electronic Journal of Qualitative Theory of Differential Equations)

   We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov–Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable. 

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4. Finite element approximation of steady flows of colloidal solutions

   https://doi.org/10.1051/m2an/2021043  

   Bonito, Andrea; Girault, Vivette; Guignard, Diane; Rajagopal, Kumbakonam R.; Süli, Endre (September 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method. 

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5. Graphical solutions to one-phase free boundary problems

   https://doi.org/10.1515/crelle-2023-0067  

   Engelstein, Max; Fernández-Real, Xavier; Yu, Hui (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein’s problem for minimal surfaces.As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity. 

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