Rough sound waves in 3D compressible Euler flow with vorticity

Disconzi, Marcelo M.; Luo, Chenyun; Mazzone, Giusy; Speck, Jared

doi:10.1007/s00029-021-00733-3

Cited by 11 publications

(12 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we can permute the vectorfield operators on the left-hand sides of (8-19a)-(8-25c) up to error terms of L ∞ size O( ε1/2 ), (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27) and on the left-hand side of (8-26) up to error terms of L ∞ size O( ε). Proof.…”

Section: Estimates For the Geometric Quantities Associated To The Aco...mentioning

confidence: 99%

“…Similarly, (8-25a) follows from commuting the transport equation (2-41) with P X X . To complete the proof, it only remains for us to prove (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) and (8-28) (with the help of the already established bounds (8-19a)-(8-25c) and (8-27)); for if ε is sufficiently small, this yields a strict improvement of the new bootstrap assumption mentioned at the beginning of the proof, and the conclusions of the proposition then follow from a standard continuity argument. We start by noting that the bounds in (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) for the pure F u -tangential derivatives of (Ω, S) are included in the bootstrap assumptions (6-6)-(6-7), as are the bounds…”

Section: Estimates For the Geometric Quantities Associated To The Aco...mentioning

confidence: 99%

“…25 However, we can prove a better result: we can show that the blowup-rates are not dominated by the toporder elliptic estimates for the transport variables, but rather by the blowup-rates for the wave variables. 26 The key to showing this is to replace the estimate (1-11) with a related L 2 estimate that features weights in the eikonal function u; see Proposition 11.4. Thanks to the u weights, the corresponding constant A in this analog of (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) can be chosen to be arbitrarily small, and thus the main contribution to the blowup-rate comes from the wave variables error terms, which are present in the "• • • " on the right-hand side of (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

Luk,

Speck

2024

Analysis & PDE

Self Cite

View full text Add to dashboard Cite

Consider a one-dimensional simple small-amplitude solution (ϱ (bkg) , v 1 (bkg) ) to the isentropic compressible Euler equations which has smooth initial data, coincides with a constant state outside a compact set, and forms a shock in finite time. Viewing (ϱ (bkg) , v 1 (bkg) ) as a plane-symmetric solution to the full compressible Euler equations in three dimensions, we prove that the shock-formation mechanism for the solution (ϱ (bkg) , v 1 (bkg) ) is stable against all sufficiently small and compactly supported perturbations. In particular, these perturbations are allowed to break the symmetry and have nontrivial vorticity and variable entropy.Our approach reveals the full structure of the set of blowup-points at the first singular time: within the constant-time hypersurface of first blowup, the solution's first-order Cartesian coordinate partial derivatives blow up precisely on the zero level set of a function that measures the inverse foliation density of a family of characteristic hypersurfaces. Moreover, relative to a set of geometric coordinates constructed out of an acoustic eikonal function, the fluid solution and the inverse foliation density function remain smooth up to the shock; the blowup of the solution's Cartesian coordinate partial derivatives is caused by a degeneracy between the geometric and Cartesian coordinates, signified by the vanishing of the inverse foliation density (i.e., the intersection of the characteristics).

show abstract

Section: Estimates For the Geometric Quantities Associated To The Aco...mentioning

confidence: 99%

Section: Estimates For the Geometric Quantities Associated To The Aco...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

Luk,

Speck

2024

Analysis & PDE

Self Cite

View full text Add to dashboard Cite

show abstract

“…The analysis in [103] relies on the new formulation of the flow presented in theorem 4.1, which allows one to employ analytical techniques based on nonlinear geometric optics and Strichartz estimates. The techniques used in [103] have roots in the works [41,100] (see also [104][105][106]) on low-regularity well-posedness for the non-relativistic compressible Euler equations, which in turn have roots in the works [56-61, 87, 99] on lowregularity well-posedness for quasilinear wave equations.…”

Section: Proposition 23 (Local Well-posedness and Continuation Princi...mentioning

confidence: 99%

The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in 1D and multi-dimensions

Abbrescia,

Speck

2023

Class. Quantum Grav.

Self Cite

View full text Add to dashboard Cite

In this article, we provide notes that complement the lectures on the relativistic Euler equations and shocks that were given by the second author at the program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime, which was hosted by the Erwin Schr¨odinger International Institute for Mathematics and Physics in Vienna in February, 2022. We set the stage by introducing a standard first-order formulation of the relativistic Euler equations and providing a brief overview of local well-posedness in Sobolev spaces. Then, using Riemann invariants, we provide the first detailed construction of a localized subset of the maximal globally hyperbolic developments of an open set of initially smooth, shock-forming isentropic solutions in 1D, with a focus on describing the singular boundary and the Cauchy horizon that emerges from the singularity. Next, we provide an overview of the new second-order formulation of the 3D relativistic Euler equations derived in [42], its rich geometric and analytic structures, their implications for the mathematical theory of shock waves, and their connection to the setup we use in our 1D analysis of shocks. We then highlight some key prior results on the study of shock formation and related problems. Furthermore, we provide an overview of how the formulation of the flow derived in [42] can be used to study shock formation in multiple spatial dimensions. Finally, we discuss various open problems tied to shocks.

show abstract

“…Their work was motivated by Christodoulou [4] and Christodoulou-Miao [8], and one of their goals to introduce the new wave equation formulation is to prove the shock formation result for the 3D compressible Euler equations. It also turns out that the system they derived is also very useful in the study of local wellposdeness; see [12,50].…”

Section: Corollary 12mentioning

confidence: 99%

Nontrivial global solutions to some quasilinear wave equations in three space dimensions

Yu¹

2022

Preprint

View full text Add to dashboard Cite

We construct nontrivial global solutions to a certain system of quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. Our construction applies to two types of equations which are of great interest. One is John's counterexamples u = u 2 t or u = u t ∆u, and the other is the 3D compressible Euler equations with no vorticity.

show abstract

Rough sound waves in 3D compressible Euler flow with vorticity

Cited by 11 publications

References 39 publications

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in 1D and multi-dimensions

Nontrivial global solutions to some quasilinear wave equations in three space dimensions

Contact Info

Product

Resources

About