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1. Existence of global weak solutions of $ p $-Navier-Stokes equations

   https://doi.org/10.3934/dcdsb.2021051  

   Liu, Jian-Guo; Zhang, Zhaoyun (January 2022, Discrete & Continuous Dynamical Systems - B)

   This paper investigates the global existence of weak solutions for the incompressible \begin{document}$ p $$\end{document}-Navier-Stokes equations in \begin{document}$$ \mathbb{R}^d $$\end{document} \begin{document}$$ (2\leq d\leq p) $$\end{document}. The \begin{document}$$ p $$\end{document}-Navier-Stokes equations are obtained by adding viscosity term to the \begin{document}$$ p $$\end{document}-Euler equations. The diffusion added is represented by the \begin{document}$$ p $$\end{document}-Laplacian of velocity and the \begin{document}$$ p $$\end{document}-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\begin{document}$$ p $$\end{document}$ distances with constraint density to be characteristic functions. 

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2. Multi-level loop equations for $$\beta $$-corners processes

   https://doi.org/10.1007/s00029-024-01006-5  

   Dimitrov, Evgeni; Knizel, Alisa (February 2025, Selecta Mathematica)

   Abstract The goal of the paper is to introduce a new set of tools for the study of discrete and continuous$$\beta $$ $\beta$-corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger–Dyson equations) for$$\beta $$ $\beta$-log gases obtained by Borot and Guionnet in (Commun. Math. Phys. 317, 447–483, 2013). In the discrete setting, our work provides a multi-level extension of the loop equations (also called Nekrasov equations) for discrete$$\beta $$ $\beta$-ensembles obtained by Borodin, Gorin and Guionnet in (Publications mathématiques de l’IHÉS 125, 1–78, 2017). 

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3. Second order $$L_p$$ estimates for subsolutions of fully nonlinear equations

   https://doi.org/10.1007/s00526-025-02929-3  

   Dong, Hongjie; Kitano, Shuhei (February 2025, Calculus of Variations and Partial Differential Equations)

   Abstract We obtain new$$L_p$$ $L_p$estimates for subsolutions to fully nonlinear equations. Based on our$$L_p$$ $L_p$estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations, critical exponents of$$L_p$$ $L_p$estimates and maximum principles, and the existence and uniqueness of solutions to fully nonlinear equations on the torus with free terms in the$$L_p$$ $L_p$spaces or in the space of Radon measures. 

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4. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

   https://doi.org/10.3934/krm.2022012  

   Nazarov, Fedor; Zumbrun, Kevin (January 2022, Kinetic and Related Models)

   We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$$ (0, +\infty) $$\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $$\end{document} stable manifolds of such equations, showing that \begin{document}$$ L^2_{loc} $$\end{document} solutions that remain sufficiently small in \begin{document}$$ L^\infty $$\end{document} (i) decay exponentially, and (ii) are \begin{document}$$ C^\infty $$\end{document} for \begin{document}$$ t>0 $$\end{document}, hence lie eventually in the \begin{document}$$ H^1 $$\end{document}$ stable manifold constructed by Pogan and Zumbrun. 

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5. Solutions of the Ginzburg–Landau equations with vorticity concentrating near a nondegenerate geodesic

   https://doi.org/10.4171/JEMS/1397  

   Colinet, Andrew; Jerrard, Robert; Sternberg, Peter (March 2025, Journal of the European Mathematical Society)

   It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg–Landau equations-\Delta u_{\varepsilon} +\varepsilon^{-2}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon} = 0, the energy and vorticity concentrate as\varepsilon\to 0around a codimension2stationary varifold – a (measure-theoretic) minimal surface. Much less is known about the question of whether, given a codimension2minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen–Cahn equation, and for the Ginzburg–Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open. We consider this question on a3-dimensional closed Riemannian manifold(M,g), and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/ vorticity concentration set of a sequence of solutions of the Ginzburg–Landau equations. 

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