Shallow water equations: viscous solutions and inviscid limit

Perepelitsa, Mikhail

doi:10.1007/s00033-012-0209-9

Cited by 12 publications

(8 citation statements)

References 23 publications

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“…Another different proof is given by LeFloch-Westdickenberg [58] for 1 < γ ≤ 5 3 . The inviscid limit of the viscous shallow water equations to the Saint-Venant system has also been established in Chen-Perepelitsa [19].…”

Section: Navier-stokes Equations: Inviscid Limitmentioning

confidence: 99%

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Weak continuity and compactness for nonlinear partial differential equations

Chen

2015

Chin. Ann. Math. Ser. B

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This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multidimensional steady compressible fluids. The weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.

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Section: Navier-stokes Equations: Inviscid Limitmentioning

confidence: 99%

“…3 . The inviscid limit of the viscous shallow water equations to the Saint-Venant system has also been established in Chen-Perepelitsa [19].…”

Section: 16)mentioning

confidence: 99%

Weak continuity and compactness for nonlinear partial differential equations

Chen

2015

Chin. Ann. Math. Ser. B

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“…In order to make our estimate, we will use the viscous shallow water equations, as given in [72] in one dimension…”

Section: Viscositymentioning

confidence: 99%

Random focusing of tsunami waves

et al. 2015

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Branched flow is a universal phenomenon of random focusing that occurs in wave or particle flows that propagate in weakly scattering, correlated random media. The consecutive effect of small random forces leads to regions of strong focusing which have the appearance of branches and originate from the formation of random caustics. This phenomenon has been experimentally and theoretically studied in various systems, ranging from experimental observations in electronic microdevices on the micrometer scale to theoretical predictions for the propagation of sound waves in the ocean, on the scale of thousands of kilometers. Reconstructions of the tsunami of March 2011 exhibited strong fluctuations in the tsunami height, associated with a filamentation of the flow, reminiscent of the structures observed for example in electron flows in semiconductor microstructures. This raises the question, to what extent are the same mechanisms at play in these very different physical systems and what impact do they have for tsunami predictions. Developing a theory of random caustics and branching in tsunami waves is the main purpose of this thesis.We will start by showing that tsunamis indeed exhibit strong focusing even when propagating over a weakly scattering region of the ocean floor. We will therefore develop the stochastic theory for the characteristic length scale on which random caustics appear in the propagations of tsunamis described by ray equations. We then confirm that the focusing regions of tsunami waves follow the scaling predicted by stochastic ray dynamics with respect to the parameters of the bathymetry. We thus show that tsunamis are indeed subject to the phenomenon of branched flow. We will furthermore demonstrate that, due to the fact that already tiny bathymetry fluctuations can be a source of branched flow, bathymetry has a severe impact on the predictability of tsunami heights. Small uncertainties in the knowledge of the ocean's bathymetry can lead to drastically wrong predictions.Because the ocean floor bathymetry is known to exhibit anisotropies and to be correlated on several length scales due to the various geological processes contributing to its formation, we later extend the general theory of branched flows to systems where the random medium is correlated on more than one single length-scale, both for tsunami waves and Hamiltonian rays, as it is also relevant to many other systems. We calculate how such correlations affect the typical length scale of branching. Our theory is then applicable to a large variety of correlation functions, either anisotropic or isotropic with multiple correlation lengths. We conclude with a proposal for an experiment which scales a tsunami event down to the size of a tank in a laboratory to study the focusing effect of bathymetry structures. Such a tool could be useful in tsunami studies and forecasting and it would allow us to experimentally verify our theoretical and numerical results on random focusing of tsunami waves.3

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“…Problems of the form (1.1) are of interest in many physical situations. For instance, they appear in the viscous shallow water problem [5], and also in the equations of gas dynamics for viscous heat conducting fluid in eulerian coordinates [18, p.256]. Apart from the physical point of view, the study of viscous IBVP along with the convergence results play an important role in the numerical analysis of hyperbolic conservation laws.…”

Section: Introductionmentioning

confidence: 99%