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1. Elastic scattering from rough surfaces in three dimensions

   https://doi.org/10.1016/j.jde.2020.03.022  

   Hu, G.; Li, P.; Zhao, Y. (January 2020, Journal of differential equations)

   Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly Lipschitz functions. In the case of flat surfaces with local perturbations, an equivalent variational formulation is deduced in a truncated bounded domain and the existence of solutions are shown for general incoming waves. The main ingredient of the proof is the radiating behavior of the Green tensor to the first boundary value problem of the Navier equation in a half-space. 

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2. NEUMANN PROBLEMS FOR p-HARMONIC FUNCTIONS, AND INDUCED NONLOCAL OPERATORS IN METRIC MEASURE SPACES

   Capogna, Luca; Kline, Joshua; Korte, Riikka; Shanmugalingam, Nageswari; Snipes, Marie (April 2022, Arxiv)

   Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆p)θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non- homogenous non-local equation (−∆p)θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality. 

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3. Global regularity for critical SQG in bounded domains

   https://doi.org/10.1002/cpa.22221  

   Constantin, Peter; Ignatova, Mihaela; Nguyen, Quoc‐Hung (January 2025, Communications on Pure and Applied Mathematics)

   Abstract We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system. 

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4. 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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5. Sharp Hadamard Local Well-Posedness, Enhanced Uniqueness and Pointwise Continuation Criterion for the Incompressible Free Boundary Euler Equations

   https://doi.org/10.1007/s40818-025-00204-4  

   Ifrim, Mihaela; Pineau, Ben; Tataru, Daniel; Taylor, Mitchell A (June 2025, Annals of PDE)

   Abstract We provide a complete local well-posedness theory inH^sbased Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the$$C^{1,\frac{1}{2}}$$regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in$$L_T^1W^{1,\infty}$$and the free surface is in$$L_T^1C^{1,\frac{1}{2}}$$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains. 

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