The stability of compressible vortex sheets in two space dimensions

Coulombel, Jean-François; Secchi, Paolo

doi:10.1512/iumj.2004.53.2526

Cited by 128 publications

(241 citation statements)

References 22 publications

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“…Such problems have been extensively studied for semilinear equations (see for example [4], [31]). Our result can be formally compared to the works of Majda on the propagation of shocks [29], [30] for systems of conservation laws and the subsequent [2], [32], [33], [14]. In these works, a short time existence, uniqueness and regularity result was established for initial data with a jump discontinuity across a surface and a precise description of the propagation of the singularity was also given.…”

Section: Remark 2 (Uniqueness)mentioning

confidence: 75%

“…However, contrary to [29], [30] the singularity that is considered in the present paper is not a shock, as it propagates along the characteristics. Moreover, unlike in [29], [30], [2], [32], [33], [14], where the problem is reformulated as an initial-boundary value problem and uniqueness is known only within the class of piecewise smooth solutions, our solution is also unique among limits of smooth solutions. In order to achieve this, the special structure of the Einstein equations in the double null foliation gauge has been heavily exploited.…”

Section: Remark 2 (Uniqueness)mentioning

confidence: 99%

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Local Propagation of Impulsive GravitationalWaves

Luk

Rodnianski²

2014

Comm Pure Appl Math

138

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Abstract. In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of non-regular characteristic data.

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Section: Remark 2 (Uniqueness)mentioning

confidence: 75%

Section: Remark 2 (Uniqueness)mentioning

confidence: 99%

Local Propagation of Impulsive GravitationalWaves

Luk

Rodnianski²

2014

Comm Pure Appl Math

138

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“…Coulombel and Secchi [4,5,6] have proved in various settings that an estimate with a loss of one derivative does hold. However, this loss of derivative creates difficulties when applying it to non-linear problems, where we have to iterate linear problems.…”

Section: Filtering Systems Of Wr Classmentioning

confidence: 99%

“…Our expectation is to find an ǫ ∈ R such that both (12,14) and (15) be strongly stable BVPs. For (5,6), this means the (UKL) condition. Since P is a homogeneous first order ΨDO, of the form ∂ d + Q(∂ t , ∇ y ), the boundary is non-characteristic; we thus ask that the Cauchy problem be well-posed for P , and that the waves be outgoing, which implies that the BVP with no boundary condition is strongly well-posed.…”

Section: Bvps For the Wave Equationmentioning

confidence: 99%

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Solvability of Hyperbolic IBVPs Through Filtering

Serre¹

2005

Methods and Applications of Analysis

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To Joel Smoller, with admiration.Abstract. In a recent paper in collaboration with S. Benzoni, F. Rousset and K. Zumbrun, we uncovered the amazing fact that besides the classes of Hadamard unstable (no estimate at all) and of strongly stable hyperbolic IBVPs (estimates without loss of derivatives), there is a third "open" class, in the sense that it is stable under small disturbances of the coefficients (we apologize for the use of the word "stable" with two different meanings) in the PDEs and in the boundary conditions. We called the latter class (WR), because it has a Weak stability property (estimates with loss of one derivative) and because it is characterized by a "Real" characteristic set for the Lopatinsliȋ determinant.We show here that with an appropriate filtering, systems in the class WR can be recast as strongly stable ones. It is remarkable that the same filtering is applied to both the solution and the data. Thus there is a hope to apply this linear theory to fixed point methods in non-linear problems, like the stability of two-dimensional vortex sheets when the jump in the velocity is larger than 2 √ 2 times the sound speed.

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Nonlinear Stability of the Current‐Vortex Sheet to the Incompressible MHD Equations

Sun

Wang

Zhang

2017

Comm Pure Appl Math

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In this paper, we solve a long-standing open problem: nonlinear stability of the current-vortex sheet in the ideal incompressible magnetohydrodynamics under the Syrovatskij stability condition. This result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instability. © 2017 Wiley Periodicals, Inc. the solution u˙WD uj ṫ ; h˙WD hj ṫ ; p˙WD pj ṫ ;

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The stability of compressible vortex sheets in two space dimensions

Cited by 128 publications

References 22 publications

Local Propagation of Impulsive GravitationalWaves

Local Propagation of Impulsive GravitationalWaves

Solvability of Hyperbolic IBVPs Through Filtering

Nonlinear Stability of the Current‐Vortex Sheet to the Incompressible MHD Equations

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