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1. Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow

   Chen, G-Q; Feldman, M; Hu, J; Xiang, W. (January 2020, SIAM journal on mathematical analysis)

   We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the nonexistence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties. 

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2. On classical de Sitter solutions and parametric control

   https://doi.org/10.1007/JHEP06(2024)101  

   Andriot, David; Ruehle, Fabian (June 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> Finding string backgrounds with de Sitter spacetime, where all approximations and corrections are controlled, is an open problem. We revisit the search for de Sitter solutions in the classical regime for specific type IIB supergravity compactifications on group manifolds, an under-explored corner of the landscape that offers an interesting testing ground for swampland conjectures. While the supergravity de Sitter solutions we obtain numerically are ambiguous in terms of their classicality, we find an analytic scaling that makes four out of six compactification radii, as well as the overall volume, arbitrarily large. This potentially provides parametric control over corrections. If we could show that these solutions, or others to be found, are fully classical, they would constitute a counterexample to conjectures stating that asymptotic de Sitter solutions do not exist. We discuss this point in great detail. 

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3. Modulation theory for soliton resonance and Mach reflection

   https://doi.org/10.1098/rspa.2021.0823  

   Ryskamp, Samuel J.; Hoefer, Mark A.; Biondini, Gino (March 2022, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Resonant Y-shaped soliton solutions to the Kadomtsev–Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully two-dimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine–Hugoniot jump conditions for the one-dimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, V-shaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation. 

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4. On backflow associated with oceanic and continental subduction

   https://doi.org/10.1093/gji/ggab246  

   Moulas, Evangelos; Brandon, Mark T; Vaughan Hammon, Joshua D; Schmalholz, Stefan M (July 2021, Geophysical Journal International)

   SUMMARY A popular idea is that accretion of sediment at a subduction zone commonly leads to the formation of a subduction channel, which is envisioned as a narrow zone located above a subducting plate and filled with vigorously circulating accreted sediment and exotic blocks. The circulation can be viewed as a forced convection, with downward flow in the lower part of the channel due to entrainment by the subducting plate, and a ‘backflow’ in the upper part of the channel. The backflow is often cited as an explanation for the exhumation of high-pressure/low-temperature metamorphic rocks from depths of 30 to 50 km. Previous analyses of this problem have mainly focused on the restricted case where the walls bounding the flow are artificially held fixed and rigid. A key question is if this configuration can be sustained on a geologically relevant timescale. We address this question using a coupled pair of corner flows. The pro-corner accounts for accretion and deformation directly above the subducting plate, and the retro-corner corresponds to a deformable region in the overlying plate. The two corners share a medial boundary, which is fully coupled but is otherwise free to rotate and deform. Our results indicate that the maintenance of a stable circulating flow in a narrow pro-corner (<15°) requires an unusually large viscosity ratio, μretro/μpro > 103. For lower viscosity ratios, the medial boundary would rotate rearwards, converting the initially narrow pro-corner into an obtuse geometry. For a stable narrow corner, we show that the backflow within the corner is caused by downward convergence of the incoming flow and an associated downward increase in dynamic pressure, which reaches a maximum at the corner point. The total pressure is thus expected to be much greater than predicted using a lithostatic gradient, which means that estimates of depth from metamorphic pressure would have to be adjusted accordingly. In addition, we show that the velocity fields associated with a forced corner flow and a buoyancy-assisted channel flow are nearly identical. As such, structural geology studies are not sufficient to distinguish between these two processes. 

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5. Evolution of truncated and bent gravity wave solitons: the Mach expansion problem

   https://doi.org/10.1017/jfm.2020.952  

   Ryskamp, Samuel; Maiden, Michelle D.; Biondini, Gino; Hoefer, Mark A. (February 2021, Journal of Fluid Mechanics)

   null (Ed.)

   The dynamics of initially truncated and bent line solitons for the Kadomtsev–Petviashvili (KPII) equation modelling internal and surface gravity waves is analysed using modulation theory. In contrast to previous studies on obliquely interacting solitons that develop from acute incidence angles, this work focuses on initial value problems for the obtuse incidence of two or three partial line solitons, which propagate away from one another. Despite counterpropagation, significant residual soliton interactions are observed with novel physical consequences. The initial value problem for a truncated line soliton – describing the emergence of a quasi-one-dimensional soliton from a wide channel – is shown to be related to the interaction of oblique solitons. Analytical descriptions for the development of weak and strong interactions are obtained in terms of interacting simple wave solutions of modulation equations for the local soliton amplitude and slope. In the weak interaction case, the long-time evolution of truncated and large obtuse angle solitons exhibits a decaying, parabolic wave profile with temporally increasing focal length that asymptotes to a cylindrical Korteweg–de Vries soliton. In contrast, the strong interaction case of slightly obtuse interacting solitons evolves into a steady, one-dimensional line soliton with amplitude reduced by an amount proportional to the incidence slope. This strong interaction is identified with the ‘Mach expansion’ of a soliton with an expansive corner, contrasting with the well-known Mach reflection of a soliton with a compressive corner. Interestingly, the critical angles for Mach expansion and reflection are the same. Numerical simulations of the KPII equation quantitatively support the analytical findings. 

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