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1. Regularity and nondegeneracy for nonlocal Bernoulli problems with variable kernels

   https://doi.org/10.1090/tran/9343  

   Snelson, Stanley; Teixeira, Eduardo (June 2025, Transactions of the American Mathematical Society)

   We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order $2s\in <\#comment/>(0,2)$. Because of the lack of translation invariance, the Caffarelli-Silvestre extension is unavailable, and we must work with the nonlocal problem directly instead of transforming to a thin free boundary problem. We prove global Hölder continuity of minimizers for both the one- and two-phase problems. Next, for the one-phase problem, we show Hölder continuity at the free boundary with the optimal exponent $s$. We also prove matching nondegeneracy estimates. A key novelty of our work is that all our findings hold without requiring any regularity assumptions on the kernel of the nonlocal operator. This characteristic makes them crucial in the development of a universal regularity theory for nonlocal free boundary problems. 

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2. Rough solutions of the relativistic Euler equations

   https://doi.org/10.1142/S0219891624500127  

   Yu, Sifan (June 2024, Journal of Hyperbolic Differential Equations)

   We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory. 

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3. A Hidden Convexity of Nonlinear Elasticity

   https://doi.org/10.1007/s10659-024-10081-w  

   Singh, Siddharth; Ginster, Janusz; Acharya, Amit (October 2024, Journal of Elasticity)

   Abstract A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response. 

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4. Well-posedness and regularity for a polyconvex energy

   https://doi.org/10.1051/cocv/2023041  

   Gangbo, Wilfrid; Jacobs, Matt; Kim, Inwon (January 2023, ESAIM: Control, Optimisation and Calculus of Variations)

   We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to theH¹-projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to constructL^r-solutions of the Navier–Stokes equation (NSE) for a short time interval. 

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5. The linearized 3d Euler equations with inflow, outflow

   https://doi.org/10.57262/ade028-0506-373  

   Gie, Gung-Min; Kelliher, James P.; Mazzucato, Anna L. (May 2023, Advances in Differential Equations)

   In 1983, Antontsev, Kazhikhov, and Monakhov published a proof of the existence and uniqueness of solutions to the 3D Euler equations in which on certain inflow boundary components fluid is forced into the domain while on other outflow components fluid is drawn out of the domain. A key tool they used was the linearized Euler equations in vorticity form. We extend their result on the linearized problem to multiply connected domains and establish compatibility conditions on the initial data that allow higher regularity solutions. 

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