On the Existence of a Maximal Cauchy Development for the Einstein Equations: a Dezornification

Sbierski, Jan

doi:10.1007/s00023-015-0401-5

Cited by 53 publications

(62 citation statements)

References 9 publications

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“…12 By this, we mean solutions that are independent of x 2 and that are constant along a family of null hyperplanes. 13 Roughly, the maximal development is the largest possible solution that is uniquely determined by the data; see, for example, [56,64] for further discussion. 14 There is precisely one exceptional equation of state for the irrotational relativistic Euler equations to which our results do not apply.…”

Section: Remark 17 (Extending the Results To The Irrotational Euler mentioning

confidence: 99%

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Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

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In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

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Section: Remark 17 (Extending the Results To The Irrotational Euler mentioning

confidence: 99%

“…In principle, the functions f could always be chosen to be polynomials with positive coefficients or exponential functions. 56 However, to avoid lengthening the paper, we typically do not specify the form of f . Throughout the rest of the paper, we make the following relative smallness assumptions.…”

Section: Smallness Assumptionsmentioning

confidence: 99%

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

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“…(65) follows from (63) and simple computations. To prove (66), we contract S α against equation (30) and use equation (29). (67) then follows from (52) and (66).…”

Section: Preliminary Identitiesmentioning

confidence: 99%

“…Next, using the identity (28), (29), and (50), we express the third-from-last product on RHS (130) as follows:…”

Section: The Transport-div-curl System For the Vorticitymentioning

confidence: 99%

“…We stress that [4] was the first work that provided sharp information about the boundary of the maximal development in more than one spatial dimension in the context of shock formation. Roughly, the maximal development is the largest possible classical solution that is uniquely determined by the initial data; see [29,41] for further discussion.…”

Section: Introductionmentioning

confidence: 99%

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The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

Disconzi

Speck

2019

Ann. Henri Poincaré

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We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and transport-div-curl equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.

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Four Lectures on Asymptotically Flat Riemannian Manifolds

Carlotto¹

2019

Einstein Equations: Physical and Mathematical Aspects of General Relativity

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On the Existence of a Maximal Cauchy Development for the Einstein Equations: a Dezornification

Cited by 53 publications

References 9 publications

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

Four Lectures on Asymptotically Flat Riemannian Manifolds

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