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1. Holographic transport beyond the supergravity approximation

   https://doi.org/10.1007/JHEP04(2024)032  

   Buchel, Alex; Cremonini, Sera; Early, Laura (April 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We set up a unified framework to efficiently compute the shear and bulk viscosities of strongly coupled gauge theories with gravitational holographic duals involving higher derivative corrections. We consider both Weyl⁴corrections, encoding the finite ’t Hooft coupling corrections of the boundary theory, and Riemann²corrections, responsible for non-equal central chargesc≠aof the theory at the ultraviolet fixed point. Our expressions for the viscosities in higher derivative holographic models are extracted from a radially conserved current and depend only on the horizon data. 

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2. Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations ^*

   https://doi.org/10.1088/1361-6544/ad7f18  

   Wu, Jiahong; Yamazaki, Kazuo (October 2024, Nonlinearity)

   Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations. 

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3. On properties of univariate max functions at local maximizers

   https://doi.org/10.1007/s11590-022-01872-y  

   Mitchell, Tim; Overton, Michael L. (March 2022, Optimization Letters)

   Abstract More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that is useful in several applications that we describe. We also investigate whether this smoothness property extends to max functions more generally. We show that the pointwise maximum of a finite set ofq-times continuously differentiable univariate functions must have zero derivative at a maximizer for$$q=1$$ $q=1$, but arbitrarily close to the maximizer, the derivative may not be defined, even when$$q=3$$ $q=3$and the maximizer is isolated. 

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4. Exact shape derivatives with unfitted finite element methods

   https://doi.org/10.1515/jnma-2024-0113  

   Shahan, Jeremy T; Walker, Shawn W (August 2025, Journal of Numerical Mathematics)

   Abstract We present an approach to shape optimization problems that uses an unfitted finite element method (FEM). The domain geometry is represented, and optimized, using a (discrete) level set function and we consider objective functionals that are defined over bulk domains. For a discrete objective functional, defined in the unfitted FEM framework, we show that theexactdiscrete shape derivative essentially matches the shape derivative at the continuous level. In other words, our approach has the benefits of both optimize-then-discretize and discretize-then-optimize approaches. Specifically, we establish the shape Fréchet differentiability of discrete (unfitted) bulk shape functionals using both the perturbation of the identity approach and direct perturbation of the level set representation. The latter approach is especially convenient for optimizing with respect to level set functions. Moreover, our Fréchet differentiability results hold foranypolynomial degree used for the discrete level set representation of the domain. We illustrate our results with some numerical accuracy tests, a simple model (geometric) problem with known exact solution, as well as shape optimization of structural designs. 

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5. Interior Regularity for Two-Dimensional Stationary Q-Valued Maps

   https://doi.org/10.1007/s00205-024-02011-w  

   Hirsch, Jonas; Spolaor, Luca (July 2024, Archive for Rational Mechanics and Analysis)

   Abstract We prove that 2-dimensionalQ-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions. 

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