Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases

Lei, Zhen; Wei, Changhua

doi:10.1007/s00208-016-1396-z

Cited by 24 publications

(16 citation statements)

References 43 publications

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“…Based on this fact, Kong, Liu and Wang [14] proved the global stability of two dimensional isentropic Chaplygin gases without vorticity by the potential theory. Lei and Wei in [21] 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 region, they showed the global radial solutions of 3D nonisentropic relativistic Chaplygin gases. For the study of relativistic membrane in general curved background manifold, only a few results are available in the literature.…”

Section: Background Materials and Outline Of This Papermentioning

confidence: 99%

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

LeFloch¹,

Wei²

2018

Journal of Differential Equations

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We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (possibly curved) Lorentzian manifold, when the spacetime metric is a perturbation of the Minkowski metric. Relying on the Hyperboloidal Foliation Method introduced by LeFloch and Ma in 2014, we revisit a theorem established earlier by Lindblad (who treated membranes in the flat Minkowski spacetime) and we provide a simpler proof of existence, which is also valid in a curved spacetime and, most importantly, leads to the important property that the total energy of the membrane is globally bounded in time. 1Our proof uses LeFloch and Ma's Hyperboloidal Foliation Method [15]- [18], which allows one to treat coupled systems of wave and Klein-Gordon equations. This method was built upon earlier work by Klainerman [12] and Hormander [9] on the quasilinear Klein-Gordon equation. The use of the hyperboloidal foliation of Minkowski spacetime to study coupled systems of wave and Klein-Gordon equation was investigated in [15] and several classes of such systems were then treated by this method, including the Einstein equations of general relativity [15,17,18]. Estimates in the hyperboloidal foliation are often more precise and it was observed in [15] that, for certain classes of equations, the energy of the solution is bounded globally in time. See LeFloch-Ma's theorem in [15, Chap. 6]. Our aim in the present paper is to extend this observation to the membrane equation (1.1). Our proof will, in addition, rely on a "double null" property of the membrane equation which was first observed by Lindblad [20]. Statement of the theoremBy the property of finite speed of propagation of the relativistic membranes, we at first specify the domain of the nonvanishing functions, which is the interior of the future light cone from the point (1, 0, 0) K := {(t, x) | r < t − 1}, and the following domain limited by two hyperboloids (with s 0 < s 1 )

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Section: Background Materials and Outline Of This Papermentioning

confidence: 99%

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

LeFloch¹,

Wei²

2018

Journal of Differential Equations

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“…In this paper, we focus our interest on the following Cauchy problem of 2D relativistic membrane equation in the form of graph ψ = ψ ( x , t ), which can also characterize the motion of isentropic Chaplygin gases without vorticity in general Lorentzian space‐time by the potential theory (see Lei and Wei)

({, \begin{matrix} array D_{μ} (\frac{1}{\sqrt{1 - h}} D^{μ} ψ) = 0, \\ array t = 0 : ψ (x, 0) = m (x), ψ_{t} (x, 0) = n (x), \end{matrix})

where repeated upper and lower indices are summed over their ranges. h = − g μν ∂ μ ψ∂ ν ψ denotes the square of the enthalpy, D denotes the covariant derivative, and g μν is the reciprocal of the following metric g = ( g μν ), which can be seen as a small homogeneous and isotropic perturbations within the class of flat Friedmann‐Lemaître‐Robertson‐Walker (FLRW) metrics

g = - d t^{2} + a false(t false) true \sum_{i = 1}^{2} {false(d x_{i} false)}^{2},

and a ( t ) > 0 satisfies

(||, \frac{1}{a false(t false)} - 1) \leq \frac{δ_{1}}{{false(1 + t false)}^{γ}}, \frac{d^{κ} a false(t false)}{d t^{κ}} \leq \frac{δ_{1}}{{false(1 + t false)}^{γ + κ}},

where δ 1 is a small parameter and

γ \geq \frac{3}{2}

κ \in N^{+}

.…”

Section: Introductionmentioning

confidence: 99%

“…Based on this fact, Kong, Liu, and Wang proved the global stability of 2D isentropic Chaplygin gases without vorticity by the potential theory. 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%

“…Remark When a ( t ) = 1, Equation is nothing but the relativistic membrane equation in the Minkowski space‐time, whose global existence result has been proved by Lindblad under the assumption of small initial data with compact support. When a ( t ) = 1 and f ( t ) is a timelike hyperplane, the global existence of smooth solution to has been proved by the authors …”

Section: Introductionmentioning

confidence: 99%

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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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“…Remark 1.1. For detailed derivation of (1.4) from (1.1), one can refer to Smoller and Temple [15] or our paper [8].…”

Section: Introductionmentioning

confidence: 99%

Spreading of the free boundary of relativistic Euler equations in a vacuum

Wei

Han

2018

Mathematical Research Letters

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Thomas C. Sideris in [J. Differential Equations 257 (2014), no. 1, 1-14] showed that the diameter of a region occupied by an ideal fluid surrounded by vacuum will grow linearly in time provided the pressure is positive and there are no singularities. In this paper, we generalize this interesting result to isentropic relativistic Euler equations with pressure p = σ 2 ρ. We will show that the results obtained by Sideris still hold for relativistic fluids. Furthermore, a family of explicit spherically symmetric solutions is constructed to illustrate our result when σ = 0, which is different from Sideris's self-similar solution.

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Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases

Cited by 24 publications

References 43 publications

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

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

Spreading of the free boundary of relativistic Euler equations in a vacuum

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