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Register a new userFormation of singularity for the rotating shallow water system
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In this paper, we investigate the formation of singularity for general two dimensional and radially symmetric solutions for rotating shallow water system from different aspects. First, the formation of singularity is proved via the study for the associated moments for two dimensional solutions. For the radial symmetric solutions, the formation of singularity is established for the initial data with compact support. Finally, the global existence or formation of singularity for the radial symmetric solutions of the rotating shallow water system are analyzed in detail when the solutions are of the form with separated variables.
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We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain threshold, we prove finite-time singularity formation for the classical solutions by studying the weighted gradients of Riemann invariants and utilizing conservation of physical energy. In fact, the singularity formation will take place for a large class of ${C}^1$ initial data whose gradients and physical energy can be arbitrarily small and higher order derivatives should be large. Second, when the initial data have constant potential vorticity, global existence of small classical solutions is established via studying an equivalent form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global existence result, the smallness condition is in terms of the higher order Sobolev norms of the initial data.
We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without coriolis forces, the rotating effect causes a coupling between two parts of Hodges decomposition of the velocity vector field, additional regularity is required in order to carry out the Friedrichs regularization and compactness arguments.
In this paper, we introduce a new extended version of the shallow water equations with surface tension which is skew-symmetric with respect to the L2 scalar product and allows for large gradients of fluid height. This result is a generalization of the results published by P. Noble and J.-P. Vila in [SIAM J. Num. Anal. (2016)] and by D. Bresch, F. Couderc, P. Noble and J.P. Vila in [C.R. Acad. Sciences Paris (2016)] which are restricted to quadratic forms of the capillary energy respectively in the one dimensional and two dimensional setting.This is also an improvement of the results by J. Lallement, P. Villedieu et al. published in [AIAA Aviation Forum 2018] where the augmented version is not skew-symetric with respect to the L2 scalar product. Based on this new formulation, we propose a new numerical scheme and perform a nonlinear stability analysis.Various numerical simulations of the shallow water equations are presented to show differences between quadratic (w.r.t the gradient of the height) and general surface tension energy when high gradients of the fluid height occur.
We study local-time well-posedness and breakdown for solutions of regularized Saint-Venant equations (regularized classical shallow water equations) recently introduced by Clamond and Dutykh. The system is linearly non-dispersive, and smooth solutions conserve an $H^1$-equivalent energy. No shock discontinuities can occur, but the system is known to admit weakly singular shock-profile solutions that dissipate energy. We identify a class of small-energy smooth solutions that develop singularities in the first derivatives in finite time.
We present an extensive numerical comparison of a family of balance models appropriate to the semi-geostrophic limit of the rotating shallow water equations, and derived by variational asymptotics in Oliver (2006) for small Rossby numbers ${mathrm{Ro}}$. This family of generalized large-scale semi-geostrophic (GLSG) models contains the $L_1$-model introduced by Salmon (1983) as a special case. We use these models to produce balanced initial states for the full shallow water equations. We then numerically investigate how well these models capture the dynamics of an initially balanced shallow water flow. It is shown that, whereas the $L_1$-member of the GLSG family is able to reproduce the balanced dynamics of the full shallow water equations on time scales of ${mathcal{O}}(1/{mathrm{Ro}})$ very well, all other members develop significant unphysical high wavenumber contributions in the ageostrophic vorticity which spoil the dynamics.
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