Singularity formation for Burgers' equation with transverse viscosity

Collot, Charles; Ghoul, Tej-Eddine; Masmoudi, Nader

doi:10.24033/asens.2513

Cited by 13 publications

(21 citation statements)

References 20 publications

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“…• In sections 3 and 6, we will study the formation of shock singularities, in which the solution remains in the interior of the regime of hyperbolicity and the breakdown is precisely due to the scenario (39). • The techniques of [51], which rely only on energy estimates, Sobolev embedding, and the fractional Sobolev-Moser calculus, can be used to save half a derivative compared to proposition 2.3, that is, to prove local well-posedness for the 3D relativistic Euler equations whenever h,ů 1 ,ů 2 ,ů 3 ,s ∈ H 5/2 + (R 3 ).…”

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.

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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.

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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.

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“…To better understand what happens, we shall examine the simplest 1D inviscid Burgers model, whose welllocalized solutions are proved to become singular in finite time. It is pointed out explicitly in [18,19] that as we are approaching the blow-up point, the blow-up solution can be well modeled by a dynamically rescaled version of a fixed profile, which belongs to a countable family F of functions, and the members in F are solutions to the self-similar Burgers equation. The choice of profile only depends on the derivatives of initial data at the point that achieves the minimum negative slope.…”

Section: Introductionmentioning

confidence: 99%

“…Thus the family F of solutions to the self-similar Burgers equation plays an important role in the blow-up phenomenon of the Burgers equation. For a detailed discussion, see [18], or the toy model in appendix A.…”

Section: Introductionmentioning

confidence: 99%

“…After the asymptotic blow-up behavior of the inviscid Burgers equation was clarified systematically in [18], the self-similar Burgers profiles have been used to explore blow-up phenomena in various systems, see [10,13,30,31,36]. The modulation method that was developed in the context of nonlinear dispersive equations, is the suitable way to utilize the self-similar Burgers profiles.…”

Section: Introductionmentioning

confidence: 99%

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Shock formation for 2D Isentropic Euler equations with self-similar variables

Su¹

2023

Preprint

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We study the 2D isentropic Euler equations with the ideal gas law. We exhibit a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions are associated with non-zero vorticity at the shock and have uniform-in-time 1/3-Hölder bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. Our proof, following Buckmaster, Shkoller and Vicol [13], is based on the stable 2D self-similar Burgers profile and the modulation method.

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“…It is remarkable that backward self-similar solutions to the one-dimensional (1D) inviscid Bergers equation u t + uu x = 0 exist for −1 < α < 0 and admit the profile u(y, −1) growing as |y| → ∞ [EF09], [CGM22]. More specifically, the profile u(y, −1) is analytic if…”

Section: Introductionmentioning

confidence: 99%

Existence of Vortex Rings in Beltrami Flows

Abe

2022

Commun. Math. Phys.

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We consider (−α)-homogeneous solutions to the stationary incompressible Euler equations in R 3 \{0} for α ≥ 0 and in R 3 for α < 0. Shvydkoy (2018) demonstrated the nonexistence of (−1)-homogeneous solutions (u, p) ∈ C 1 (R 3 \{0}) and (−α)-homogeneous solutions in the range 0 ≤ α ≤ 2 for the Beltrami and axisymmetric flows. Namely, no (−α)-homogeneous solutions (u, p) ∈ C 1 (R 3 \{0}) for 1 ≤ α ≤ 2 and (u, p) ∈ C 2 (R 3 \{0}) for 0 ≤ α < 1 exist among these particular classes of flows other than irrotational solutions for integers α. The nonexistence result of the Beltrami (−α)-homogeneous solutions (u, p) ∈ C 2 (R 3 \{0}) holds for all α < 1. We show the nonexistence of axisymmetric (−α)-homogeneous solutions without swirls (u, p)The main result of this study is the existence of axisymmetric (−α)-homogeneous solutions in the complementary range α ∈ R\[0, 2]. More specifically, we show the existence of axisymmetric Beltrami (−α)homogeneous solutions (u, p) ∈ C 1 (R 3 \{0}) for α > 2 and (u, p) ∈ C(R 3 ) for α < 0 and axisymmetric (−α)homogeneous solutions with a nonconstant Bernoulli function (u, p) ∈ C 1 (R 3 \{0}) for α > 2 and (u, p) ∈ C(R 3 ) for α < −2, including axisymmetric (−α)-homogeneous solutions without swirls (u, p) ∈ C 2 (R 3 \{0}) for α > 2 and (u, p) ∈ C 1 (R 3 \{0}) ∩ C(R 3 ) for α < −2. This is the first existence result on (−α)-homogeneous solutions with no explicit forms.The level sets of the axisymmetric stream function of the irrotational (−α)-homogeneous solutions in the cross-section are the Jordan curves for α = 3. For 2 < α < 3, we show the existence of axisymmetric (−α)homogeneous solutions whose stream function level sets are the Jordan curves. They provide new examples of the Beltrami/Euler flows in R 3 \{0} whose level sets of the proportionality factor/Bernoulli surfaces are nested surfaces created by the rotation of the sign "∞".

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Singularity formation for Burgers' equation with transverse viscosity

Cited by 13 publications

References 20 publications

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

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

Shock formation for 2D Isentropic Euler equations with self-similar variables

Existence of Vortex Rings in Beltrami Flows

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