A sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates

Ettinger, Boris; Lindblad, Hans

doi:10.4007/annals.2017.185.1.6

Cited by 19 publications

(15 citation statements)

References 7 publications

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“…To prove ill-posedness for (1.1), as in [3,16,30,31], we work under planar symmetry. And the solution satisfies U(x 1 , x 2 , x 3 , t) = U (x 1 , t).…”

Section: Decomposition Of Wavesmentioning

confidence: 99%

“…Under planar symmetry, Lindblad gave sharp counterexamples to the lowregularity local well-posedness for semilinear and certain quasilinear wave equations in [29][30][31]. In [16], Ettinger-Lindblad generalized the above results to the Einstein's equations and constructed a sharp counterexample for local well-posedness of Einstein vacuum equations in wave coordinates. An exploration by Granowski [17] later showed that Lindblad's H s ill-posedness in [31] is stable under general perturbations.…”

Section: Introductionmentioning

confidence: 96%

“…Difficulties and Strategies. To prove the ill-posedness result, as in [3,16,30,31] we work under planar symmetry, the MHD system then obeys…”

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confidence: 99%

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Low regularity ill-posedness and shock formation for 3D ideal compressible MHD

An¹,

Chen²,

Yin³

2021

Preprint

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The study of magnetohydrodynamics (MHD) significantly boosts the understanding and development of solar physics, planetary dynamics and controlled nuclear fusion. Dynamical properties of the MHD system involve nonlinear interactions of waves with multiple travelling speeds (the fast and slow magnetosonic waves, the Alfvén wave and the entropy wave). One intriguing topic is the shock phenomena accompanied by the magnetic field, which have been affirmed by astronomical observations. However, permitting the residence of all above multi-speed waves, mathematically, whether one can prove shock formation for 3D MHD is still open. The multiple-speed nature of the MHD system makes it fascinating and challenging. In this paper, we give an affirmative answer to the above question. For 3D ideal compressible MHD, we construct examples of shock formation allowing the presence of all characteristic waves with multiple wave speeds. Building on our construction, we further prove that the Cauchy problem for 3D ideal MHD is H 2 ill-posed. And this is caused by the instantaneous shock formation. In particular, when the magnetic field is absent, we also provide a desired low-regularity illposedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proof for 3D MHD is based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D ideal MHD equations into a 7 × 7 non-strictly hyperbolic system. Via detailed calculations, we reveal its hidden subtle structures. With them we give a complete description of MHD dynamics up to the earliest singular event, when a shock forms.

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“…To prove ill-posedness for (1.1), as in [3,16,30,31], we work under planar symmetry. And the solution satisfies U(x 1 , x 2 , x 3 , t) = U (x 1 , t).…”

Section: Decomposition Of Wavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 1 more Smart Citation

Low regularity ill-posedness and shock formation for 3D ideal compressible MHD

An¹,

Chen²,

Yin³

2021

Preprint

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“…In addition to the stability problem of Minkowski space, spacetime harmonic gauge is also used in the context of the cosmological stability problem by [12,13]. In addition to these examples, there are numerous other studies (e.g., [14,15,16]) that utilize this particular choice of gauge. Andersson and Moncrief [18] proved an asymptotic stability result of the Milne universe utilizing the CMCSH gauge.…”

Section: Introductionmentioning

confidence: 99%

Local well-posedness of the Einstein-Yang-Mills system in CMCSHGC gauge

Mondal¹

2021

Preprint

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We study the local well-posedness of the Einstein-Yang-Mills equations in constant mean extrinsic curvature spatial harmonic generalized Coulomb gauge (CMCSHGC). In this choice of gauge, the complete Einstein-Yang-Mills equations reduce to a coupled elliptic-hyperbolic system. Utilizing the method developed by Andersson and Moncrief [1], we establish the existence of a unique, local, continuousin-time solution of this coupled system. This yields an 'in time' continuation criteria of the solutions which is to be used in the potential future proof of an improved continuation criteria for this coupled system utilizing Moncrief's light cone estimate technique.

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“…is ill-posed in H 2 (see also Lindblad [19], Ettinger and Linblad [4] for quasi-linear case). Then it is natural to expect (1.1) is ill-posed in H s l in general when 2 ≤ p ≤ 1 + 4 n−1 .…”

Section: Introductionmentioning

confidence: 99%

Concerning ill-posedness for semilinear wave equations

Liu

Wang

2018

Preprint

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In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in H s with spatial dimension n ≤ 5. We show this equation, with power 2 ≤ p ≤ 1 + 4/(n − 1), is (strongly) ill-posed in H s with s = (n + 5)/4 in general. Moreover, when the nonlinearity is quadratic we establish a characterization of the structure of nonlinear terms in terms of the regularity. As a byproduct, we give an alternative proof of the failure of the local in time endpoint scale-invariant L 4/(n−1) t L ∞x Strichartz estimates. Finally, as an application, we also prove ill-posed results for some semilinear half wave equations.

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A sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates

Cited by 19 publications

References 7 publications

Low regularity ill-posedness and shock formation for 3D ideal compressible MHD

Low regularity ill-posedness and shock formation for 3D ideal compressible MHD

Local well-posedness of the Einstein-Yang-Mills system in CMCSHGC gauge

Concerning ill-posedness for semilinear wave equations

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