Singularities of relativistic membranes

Eggers, Jens; Hoppe, Jens; Hynek, Mariusz; Suramlishvili, N.

doi:10.1515/geofl-2015-0003

Cited by 16 publications

(27 citation statements)

References 10 publications

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“…The lip describes how the overturned region (and thus the shock) spreads in space, and corresponds to similar results found in ( [17]) and ( [19]).…”

Section: Shock Conditionsupporting

confidence: 82%

“…To describe the neighborhood of the singularity, we use a self-similar description ( [16]), in analogy to caustic singularities in two dimensions ( [17]), and shocks in the dKP equation ( [19]). In the self-similar region, we assume the scalings x ∝ t β1 , y ∝ t β2 , and u ∝ t α , where x = x − x 0 , y = y − y 0 , and t = t 0 − t, so that t > 0 before the singularity, and t < 0 after.…”

Section: Similarity Structurementioning

confidence: 99%

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Spatial structure of shock formation

et al. 2017

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The formation of a singularity in a compressible gas, as described by the Euler equation, is characterized by the steepening, and eventual overturning of a wave. Using a self-similar description in two space dimensions, we show that the spatial structure of this process, which starts at a point, is equivalent to the formation of a caustic, i.e. to a cusp catastrophe. The lines along which the profile has infinite slope correspond to the caustic lines, from which we construct the position of the shock. By solving the similarity equation, we obtain a complete local description of wave steepening and of the spreading of the shock from a point

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“…The lip describes how the overturned region (and thus the shock) spreads in space, and corresponds to similar results found in ( [17]) and ( [19]).…”

Section: Shock Conditionsupporting

confidence: 82%

Section: Similarity Structurementioning

confidence: 99%

Spatial structure of shock formation

et al. 2017

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“…Inserted into (87), this yields which is precisely the solution (82) to the potential of the inviscid Burgers' equation.…”

Section: The Kinematic Wave Equationmentioning

confidence: 99%

Singularity theory of plane curves and its applications

Eggers

Suramlishvili

2017

European Journal of Mechanics - B/Fluids

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We review the classification of singularities of smooth functions from the perspective of applications in the physical sciences, restricting ourselves to functions of a real parameter t onto the plane (x, y). Singularities arise when the derivatives of x and y with respect to the parameter vanish. Near singularities the curves have a universal unfolding, described by a finite number of parameters. We emphasize the scaling properties near singularities, characterized by similarity exponents, as well as scaling functions, which describe the shape. We discuss how singularity theory can be used to find and/or classify singularities found in science and engineering, in particular as described by partial differential equations (PDE's). In the process, we point to limitations of the method, and indicate directions of future work.

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“…In this section, some important notations and lemmas are given. We first introduce the following Klainerman's vector fields Γ = (Γ 0 , … , Γ 6 ) = ( 0 , 1 , 2 , Ω 01 , Ω 02 , Ω 12 , S), (9) where Ω 01 = t 1 + x 1 t , Ω 02 = t 2 + x 2 t , Ω 12 = x 1 2 − x 2 1 , S = t t + r r .…”

Section: Preliminariesmentioning

confidence: 99%

“…Lei and Wei investigated the relationship between relativistic membrane equation and relativistic Chaplygin gases and with the help of the “null structure” of relativistic membrane equation in exterior domain; they showed the global existence of radial solutions to 3D nonisentropic relativistic Chaplygin gases. Since here, we are interested in the global stability of relativistic membrane equations, we do not list the blowup result in detail, we refer to Eggers and Hoppe() for the interested readers.…”

Section: Introductionmentioning

confidence: 99%

Global existence of smooth solution to 2D relativistic membrane equation in asymptotically flat FLRW space‐time

Wei

2019

Math Methods in App Sciences

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This paper is concerned with the 2D relativistic membrane equation evolving in curved space‐time, whose metric is prescribed as a small perturbation to the flat Friedmann‐Lemaître‐Robertson‐Walker (FLRW) metric with time‐dependent coefficients. Thanks to the partial “null structure” of the membrane equation and the properties of the background metric, we could prove the global stability of a class of time‐dependent solutions by weighted energy estimate.

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Singularities of relativistic membranes

Abstract: Pointing out a crucial relation with caustics of the eikonal equation we discuss the singularity formation of 2-dimensional surfaces that sweep out 3-manifolds of zero mean curvature in R 3,1 .

Cited by 16 publications

References 10 publications

Spatial structure of shock formation

Spatial structure of shock formation

Singularity theory of plane curves and its applications

Global existence of smooth solution to 2D relativistic membrane equation in asymptotically flat FLRW space‐time

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