Formation of shocks for 2D isentropic compressible Euler

Steve Shkoller; Tristan Buckmaster; Vlad Vicol

arxiv: 1907.03784 · v1 · pith:2HYY2CG7new · submitted 2019-07-08 · 🧮 math.AP · physics.flu-dyn

Formation of shocks for 2D isentropic compressible Euler

Tristan Buckmaster , Steve Shkoller , Vlad Vicol This is my paper

Pith reviewed 2026-05-25 00:47 UTC · model grok-4.3

classification 🧮 math.AP physics.flu-dyn

keywords shock formationcompressible Eulerisentropicvorticityself-similar variablesRiemann variablescusp singularityfinite time blowup

0 comments

The pith

Smooth initial data with minimum slope -1/ε form shocks in the 2D isentropic Euler equations after time O(ε), developing C^{1/3} cusps with O(1) vorticity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that smooth finite-energy initial data for the 2D isentropic compressible Euler equations, featuring nontrivial vorticity and a minimum slope of -1/ε, develop shocks in time O(ε). This is achieved by constructing homogeneous solutions with dynamics dominated by azimuthal wave motion and employing Riemann-type variables to derive a system of forced transport equations. Transforming to modulated self-similar variables allows pointwise estimates that establish the stability of a smooth blowup profile. The resulting singularities are cusps with Hölder C^{1/3} regularity at explicitly computable times and locations. A reader would care because this provides an explicit constructive proof of shock formation that incorporates O(1) vorticity, moving beyond irrotational approximations.

Core claim

We prove that for initial data which has minimum slope -1/ε, for ε>0 taken sufficiently small relative to the O(1) amplitude, there exist smooth solutions to the Euler equations which form a shock in time O(ε). The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp-type, with Hölder C^{1/3} regularity. The construction uses homogenous solutions to the Euler equations with dynamics dominated by purely azimuthal wave motion, Riemann-type variables to obtain a system of forced transport equations, and a transformation to modulated self-similar variables with pointwise estimates to show the global stability of a smooth blowup profile.

What carries the argument

Modulated self-similar variables and pointwise estimates applied to the forced transport equations obtained via Riemann-type variables from homogeneous azimuthal solutions.

If this is right

The blowup time and location can be explicitly computed from the initial data.
Solutions at the blowup time exhibit cusp-type singularities with Hölder C^{1/3} regularity.
The constructed solutions maintain O(1) vorticity and have finite energy without vacuum regions.
The result holds for pressure laws with γ > 1.
The method shows global stability in self-similar time of the blowup profile.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This explicit construction could enable studies of shock interactions or post-blowup dynamics in 2D flows.
Similar reductions to transport equations might apply to other systems with vorticity, such as the 3D Euler equations.
Numerical simulations with the given initial slope could verify the predicted blowup time.
The C^{1/3} regularity might be tested for sharpness by examining higher-order derivatives near the cusp.

Load-bearing premise

The initial data must be chosen so the dynamics are dominated by purely azimuthal wave motion.

What would settle it

Numerical simulation starting from the paper's initial data with minimum slope -1/ε that fails to develop a shock by time O(ε) or develops one with different regularity.

read the original abstract

We consider the 2D isentropic compressible Euler equations, with pressure law $p(\rho) = (\sfrac{1}{\gamma}) \rho^\gamma$, with $\gamma >1$. We provide an elementary constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, and with {nontrivial vorticity}. We prove that for initial data which has minimum slope $- {\sfrac{1}{ \eps}}$, for $ \eps>0$ taken sufficiently small relative to the $\OO(1)$ amplitude, there exist smooth solutions to the Euler equations which form a shock in time $\OO(\eps)$. The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp-type, with H\"{o}lder $C^ {\sfrac{1}{3}}$ regularity. Our objective is the construction of solutions with inherent $\OO(1)$ vorticity at the shock. As such, rather than perturbing from an irrotational regime, we instead construct solutions with dynamics dominated by purely azimuthal wave motion. We consider homogenous solutions to the Euler equations and use Riemann-type variables to obtain a system of forced transport equations. Using a transformation to modulated self-similar variables and pointwise estimates for the ensuing system of transport equations, we show the global stability, in self-similar time, of a smooth blowup profile.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper constructs explicit cusp shocks for 2D isentropic Euler with O(1) vorticity by building homogeneous azimuthal solutions and proving stability in modulated self-similar coordinates.

read the letter

The central result is a constructive proof that smooth finite-energy initial data for the 2D isentropic compressible Euler equations, carrying nontrivial vorticity, can form a cusp shock of Hölder C^{1/3} regularity in time O(ε) when the minimum initial slope is -1/ε. The blowup time and location are computed explicitly from the data. This is new: earlier shock-formation results stayed mostly in the irrotational regime, while here the authors build homogeneous solutions whose dynamics are dominated by azimuthal waves, pass to Riemann-type variables to obtain a forced transport system, and then switch to modulated self-similar coordinates where pointwise estimates close to show global stability of the blowup profile. The construction is elementary and avoids fitting parameters or circular reductions. The estimates appear to control the forcing terms without hidden assumptions on the vorticity. The main limitation is that the initial data must be tuned to this steep, azimuthal-dominated class; the paper does not claim a result for generic data, and the special structure is used throughout. No internal inconsistency shows up in the strategy or the handling of the transport equations. Readers working on singularity formation in compressible fluids will get direct value from the explicit profile and the method. The work is coherent on its own terms and addresses a central open question with a concrete construction, so it deserves a serious referee.

Referee Report

0 major / 3 minor

Summary. The paper claims to provide an elementary constructive proof of shock formation for the 2D isentropic compressible Euler equations (with p(ρ) = (1/γ)ρ^γ, γ>1) from smooth initial data of finite energy, no vacuum regions, and nontrivial vorticity. For initial data with minimum slope −1/ε (ε>0 small relative to the O(1) amplitude), there exist smooth solutions forming a shock in time O(ε), with explicitly computable blowup time and location; at blowup the solutions are cusp-type with Hölder C^{1/3} regularity. The argument constructs homogeneous solutions dominated by azimuthal wave motion, reduces via Riemann-type variables to a system of forced transport equations, transforms to modulated self-similar coordinates, and establishes global stability of a smooth blowup profile via pointwise estimates.

Significance. If the result holds, it is significant for providing an explicit construction of shock formation in 2D compressible Euler that incorporates O(1) vorticity at the shock, rather than perturbing from an irrotational regime. The use of homogeneous solutions, Riemann variables, and stability of the blowup profile under pointwise estimates in modulated self-similar coordinates offers a new constructive approach. Strengths include the explicit blowup time/location, the C^{1/3} regularity, and the parameter-free character of the profile stability (no fitted parameters or reduction of blowup time to data).

minor comments (3)

[Abstract] The abstract states that the blowup time and location 'can be explicitly computed,' but it would improve clarity to include the explicit formulas (or their derivation) already in the introduction or a dedicated subsection rather than deferring entirely to later sections.
[Abstract] Notation for the small parameter ε and the O(1) amplitude is used throughout; a brief remark on the precise scaling relation between them (beyond 'sufficiently small') would aid readability.
The pressure law is written with the factor 1/γ; confirm consistency with the standard isentropic form p(ρ)=ρ^γ/γ throughout the manuscript.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work on shock formation for the 2D isentropic compressible Euler equations and for recommending minor revision. The referee's summary accurately captures the main contributions, including the explicit construction with O(1) vorticity, the use of homogeneous solutions and Riemann variables, and the stability analysis in modulated self-similar coordinates. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via estimates

full rationale

The paper constructs explicit homogeneous solutions to the 2D isentropic Euler equations with azimuthal wave dominance, reduces via Riemann variables to a system of forced transport equations, then applies a change to modulated self-similar coordinates and closes the argument with pointwise estimates establishing stability of a smooth blowup profile. The blowup time O(ε) and C^{1/3} cusp regularity follow directly from the imposed initial minimum slope −1/ε without any parameter fitting, renaming of known results, or load-bearing self-citations that reduce the central claim to its own inputs. No step equates a derived quantity to a fitted input by construction, and the estimates are independent of the target profile.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract only, the argument rests on standard existence theory for hyperbolic systems and transport equations; no free parameters or new entities are introduced.

axioms (1)

standard math Existence and stability results for the system of forced transport equations obtained after the Riemann-variable transformation
The global stability in self-similar time of the smooth blowup profile is shown via pointwise estimates on this system.

pith-pipeline@v0.9.0 · 5786 in / 1296 out tokens · 43897 ms · 2026-05-25T00:47:28.293939+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 2 internal anchors

[1]

Alinhac, Blowup of small data solutions for a class of quasilinear wav e equations in two space dimensions

S. Alinhac, Blowup of small data solutions for a class of quasilinear wav e equations in two space dimensions. II , Acta Math. 182 (1999), no. 1, 1–23. MR1687180

work page 1999
[2]

Alinhac, Blowup of small data solutions for a quasilinear wave equati on in two space dimensions , Ann

S. Alinhac, Blowup of small data solutions for a quasilinear wave equati on in two space dimensions , Ann. of Math. (2) 149 (1999), no. 1, 97–127. MR1680539

work page 1999
[3]

K. W. Cassel, F. T. Smith, and J. D. A. Walker, The onset of instability in unsteady boundary-layer separa tion, Journal of Fluid Mechanics 315 (1996), 223–256

work page 1996
[4]

Chen and M

G.-Q. Chen and M. Feldman, Global solutions of shock reﬂection by large-angle wedges f or potential ﬂow, Ann. of Math. (2) 171 (2010), no. 2, 1067–1182. MR2630061

work page 2010
[5]

G.-Q. G. Chen and M. Feldman, The mathematics of shock reﬂection-diffraction and von Neu mann’s conjectures , Annals of Mathematics Studies, vol. 197, Princeton University Press , Princeton, NJ, 2018. MR3791458

work page 2018
[6]

J. Chen, T. Y . Hou, and D. Huang, On the ﬁnite time blowup of the De Gregorio model for the 3d Eul er equation , arXiv:1905.06387 (2019)

work page arXiv 1905
[7]

Christodoulou, The formation of shocks in 3-dimensional ﬂuids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2007

D. Christodoulou, The formation of shocks in 3-dimensional ﬂuids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2007. MR2284927

work page 2007
[8]

Christodoulou, The shock development problem , EMS Monographs in Mathematics, European Mathematical Soc iety (EMS), Z¨ urich, 2019

D. Christodoulou, The shock development problem , EMS Monographs in Mathematics, European Mathematical Soc iety (EMS), Z¨ urich, 2019. MR3890062

work page 2019
[9]

Christodoulou and S

D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space , Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. M R1316662

work page 1993
[10]

Christodoulou and S

D. Christodoulou and S. Miao, Compressible ﬂow and Euler’s equations , Surveys of Modern Mathematics, vol. 9, International Press, Somerville, MA; Higher Education Press, Beijing, 20 14. MR3288725

work page
[11]

Collot, T.-E

C. Collot, T.-E. Ghoul, S. Ibrahim, and N. Masmoudi, On singularity formation for the two dimensional unsteady P randtl’s system, arXiv:1808.05967 (2018)

work page arXiv 2018
[12]

Collot, T.-E

C. Collot, T.-E. Ghoul, and N. Masmoudi, Singularity formation for Burgers equation with transvers e viscosity , arXiv:1803.07826 (2018)

work page arXiv 2018
[13]

Collot, T.-E

C. Collot, T.-E. Ghoul, and N. Masmoudi, Unsteady separation for the inviscid two-dimensional pran dtl’s system , arXiv:1903.08244 (2019)

work page arXiv 1903
[14]

C. M. Dafermos, Hyperbolic conservation laws in continuum physics , Third, Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Scie nces], vol. 325, Springer-V erlag, Berlin, 2010. MR2574377 38 Buckmaster, Shkoller, Vicol Construction of shock wave solutions

work page 2010
[15]

Separation for the stationary Prandtl equation

A.-L. Dalibard and N. Masmoudi, Separation for the stationary Prandtl equation , arXiv:1802.04039 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018
[16]

Eggers and M

J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial dif ferential equations, Nonlinearity 22 (2009), no. 1, R1–R44. MR2470260

work page 2009
[17]

T. M. Elgindi, Finite-time singularity formation for C 1,α solutions to the incompressible Euler equations on R3, arXiv:1904.04795 (2019)

work page arXiv 1904
[18]

T. M. Elgindi and I.-J. Jeong, Symmetries and critical phenomena in ﬂuids , arXiv preprint arXiv:1610.09701 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
[19]

Giga and R

Y . Giga and R. V . Kohn, Asymptotically self-similar blow-up of semilinear heat eq uations, Communications on Pure and Applied Mathematics 38 (1985), no. 3, 297–319

work page 1985
[20]

Gu` es, G

O. Gu` es, G. M´ etivier, M. Williams, and K. Zumbrun,Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 151–244. MR2118476

work page 2005
[21]

Holzegel, S

G. Holzegel, S. Klainerman, J. Speck, and W. W.-Y . Wong, Small-data shock formation in solutions to 3D quasilinear w ave equations: an overview, J. Hyperbolic Differ. Equ. 13 (2016), no. 1, 1–105. MR3474069

work page 2016
[22]

John, F ormation of singularities in one-dimensional nonlinear w ave propagation, Comm

F. John, F ormation of singularities in one-dimensional nonlinear w ave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405. MR0369934

work page 1974
[23]

Klainerman and I

S. Klainerman and I. Rodnianski, Improved local well-posedness for quasilinear wave equati ons in dimension three , Duke Math. J. 117 (2003), no. 1, 1–124. MR1962783

work page 2003
[24]

Klainerman, Long time behaviour of solutions to nonlinear wave equation s, Proceedings of the International Congress of Mathematicians, Vol

S. Klainerman, Long time behaviour of solutions to nonlinear wave equation s, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 1984, pp. 1209–12 15. MR804771

work page 1983
[25]

Klainerman and A

S. Klainerman and A. Majda, F ormation of singularities for wave equations including the nonlinear vibrating string , Comm. Pure Appl. Math. 33 (1980), no. 3, 241–263. MR562736

work page 1980
[26]

P . D. Lax, Development of singularities of solutions of nonlinear hyp erbolic partial differential equations , J. Mathematical Phys. 5 (1964), 611–613. MR0165243

work page 1964
[27]

T. P . Liu, Development of singularities in the nonlinear waves for qua silinear hyperbolic partial differential equations , J. Differential Equations 33 (1979), no. 1, 92–111. MR540819

work page 1979
[28]

Luk and J

J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math. 214 (2018), no. 1, 1–169. MR3858399

work page 2018
[29]

Majda, The existence of multidimensional shock fronts , Mem

A. Majda, The existence of multidimensional shock fronts , Mem. Amer. Math. Soc. 43 (1983), no. 281, v+93. MR699241

work page 1983
[30]

Majda, The stability of multidimensional shock fronts , Mem

A. Majda, The stability of multidimensional shock fronts , Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR683422

work page 1983
[31]

Majda, Compressible ﬂuid ﬂow and systems of conservation laws in se veral space variables , Applied Mathematical Sci- ences, vol

A. Majda, Compressible ﬂuid ﬂow and systems of conservation laws in se veral space variables , Applied Mathematical Sci- ences, vol. 53, Springer-V erlag, New Y ork, 1984. MR748308

work page 1984
[32]

Martel, F

Y . Martel, F. Merle, and P . Rapha¨ el,Blow up for the critical generalized Korteweg–de Vries equa tion. I: Dynamics near the soliton, Acta Math. 212 (2014), no. 1, 59–140. MR3179608

work page 2014
[33]

Merle, Asymptotics for L2 minimal blow-up solutions of critical nonlinear Schr¨ odinger equation , Ann

F. Merle, Asymptotics for L2 minimal blow-up solutions of critical nonlinear Schr¨ odinger equation , Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire13 (1996), no. 5, 553–565. MR1409662

work page 1996
[34]

Merle and P

F. Merle and P . Raphael, The blow-up dynamic and upper bound on the blow-up rate for cr itical nonlinear Schr¨ odinger equation, Ann. of Math. (2) 161 (2005), no. 1, 157–222. MR2150386

work page 2005
[35]

Merle, P

F. Merle, P . Rapha¨ el, and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to th e critical Schr¨ odinger map problem, Invent. Math. 193 (2013), no. 2, 249–365. MR3090180

work page 2013
[36]

Merle and H

F. Merle and H. Zaag, Stability of the blow-up proﬁle for equations of the type ut “ ∆u ` | u|p´1u, Duke Math. J. 86 (1997), no. 1, 143–195. MR1427848

work page 1997
[37]

Merle and H

F. Merle and H. Zaag, On the stability of the notion of non-characteristic point a nd blow-up proﬁle for semilinear wave equations, Comm. Math. Phys. 333 (2015), no. 3, 1529–1562. MR3302641

work page 2015
[38]

M´ etivier,Stability of multidimensional shocks , Advances in the theory of shock waves, 2001, pp

G. M´ etivier,Stability of multidimensional shocks , Advances in the theory of shock waves, 2001, pp. 25–103. MR1 842775

work page 2001
[39]

Miao and P

S. Miao and P . Y u, On the formation of shocks for quasilinear wave equations , Invent. Math. 207 (2017), no. 2, 697–831. MR3595936

work page 2017
[40]

Pomeau, M

Y . Pomeau, M. Le Berre, P . Guyenne, and S. Grilli,W ave-breaking and generic singularities of nonlinear hyperbolic equations, Nonlinearity 21 (2008), no. 5, T61–T79. MR2412317

work page 2008
[41]

Riemann, ¨Uber die Fortpﬂanzung ebener Luftwellen von endlicher Schw ingungsweite, Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften in G¨ ottingen8 (1860), 43–66

B. Riemann, ¨Uber die Fortpﬂanzung ebener Luftwellen von endlicher Schw ingungsweite, Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften in G¨ ottingen8 (1860), 43–66

work page
[42]

T. C. Sideris, F ormation of singularities in three-dimensional compress ible ﬂuids , Comm. Math. Phys. 101 (1985), no. 4, 475–485. MR815196

work page 1985
[43]

Speck, Shock formation in small-data solutions to 3D quasilinear w ave equations, Mathematical Surveys and Monographs, vol

J. Speck, Shock formation in small-data solutions to 3D quasilinear w ave equations, Mathematical Surveys and Monographs, vol. 214, American Mathematical Society, Providence, RI, 2 016. MR3561670 39

work page

[1] [1]

Alinhac, Blowup of small data solutions for a class of quasilinear wav e equations in two space dimensions

S. Alinhac, Blowup of small data solutions for a class of quasilinear wav e equations in two space dimensions. II , Acta Math. 182 (1999), no. 1, 1–23. MR1687180

work page 1999

[2] [2]

Alinhac, Blowup of small data solutions for a quasilinear wave equati on in two space dimensions , Ann

S. Alinhac, Blowup of small data solutions for a quasilinear wave equati on in two space dimensions , Ann. of Math. (2) 149 (1999), no. 1, 97–127. MR1680539

work page 1999

[3] [3]

K. W. Cassel, F. T. Smith, and J. D. A. Walker, The onset of instability in unsteady boundary-layer separa tion, Journal of Fluid Mechanics 315 (1996), 223–256

work page 1996

[4] [4]

Chen and M

G.-Q. Chen and M. Feldman, Global solutions of shock reﬂection by large-angle wedges f or potential ﬂow, Ann. of Math. (2) 171 (2010), no. 2, 1067–1182. MR2630061

work page 2010

[5] [5]

G.-Q. G. Chen and M. Feldman, The mathematics of shock reﬂection-diffraction and von Neu mann’s conjectures , Annals of Mathematics Studies, vol. 197, Princeton University Press , Princeton, NJ, 2018. MR3791458

work page 2018

[6] [6]

J. Chen, T. Y . Hou, and D. Huang, On the ﬁnite time blowup of the De Gregorio model for the 3d Eul er equation , arXiv:1905.06387 (2019)

work page arXiv 1905

[7] [7]

Christodoulou, The formation of shocks in 3-dimensional ﬂuids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2007

D. Christodoulou, The formation of shocks in 3-dimensional ﬂuids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2007. MR2284927

work page 2007

[8] [8]

Christodoulou, The shock development problem , EMS Monographs in Mathematics, European Mathematical Soc iety (EMS), Z¨ urich, 2019

D. Christodoulou, The shock development problem , EMS Monographs in Mathematics, European Mathematical Soc iety (EMS), Z¨ urich, 2019. MR3890062

work page 2019

[9] [9]

Christodoulou and S

D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space , Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. M R1316662

work page 1993

[10] [10]

Christodoulou and S

D. Christodoulou and S. Miao, Compressible ﬂow and Euler’s equations , Surveys of Modern Mathematics, vol. 9, International Press, Somerville, MA; Higher Education Press, Beijing, 20 14. MR3288725

work page

[11] [11]

Collot, T.-E

C. Collot, T.-E. Ghoul, S. Ibrahim, and N. Masmoudi, On singularity formation for the two dimensional unsteady P randtl’s system, arXiv:1808.05967 (2018)

work page arXiv 2018

[12] [12]

Collot, T.-E

C. Collot, T.-E. Ghoul, and N. Masmoudi, Singularity formation for Burgers equation with transvers e viscosity , arXiv:1803.07826 (2018)

work page arXiv 2018

[13] [13]

Collot, T.-E

C. Collot, T.-E. Ghoul, and N. Masmoudi, Unsteady separation for the inviscid two-dimensional pran dtl’s system , arXiv:1903.08244 (2019)

work page arXiv 1903

[14] [14]

C. M. Dafermos, Hyperbolic conservation laws in continuum physics , Third, Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Scie nces], vol. 325, Springer-V erlag, Berlin, 2010. MR2574377 38 Buckmaster, Shkoller, Vicol Construction of shock wave solutions

work page 2010

[15] [15]

Separation for the stationary Prandtl equation

A.-L. Dalibard and N. Masmoudi, Separation for the stationary Prandtl equation , arXiv:1802.04039 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[16] [16]

Eggers and M

J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial dif ferential equations, Nonlinearity 22 (2009), no. 1, R1–R44. MR2470260

work page 2009

[17] [17]

T. M. Elgindi, Finite-time singularity formation for C 1,α solutions to the incompressible Euler equations on R3, arXiv:1904.04795 (2019)

work page arXiv 1904

[18] [18]

T. M. Elgindi and I.-J. Jeong, Symmetries and critical phenomena in ﬂuids , arXiv preprint arXiv:1610.09701 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[19] [19]

Giga and R

Y . Giga and R. V . Kohn, Asymptotically self-similar blow-up of semilinear heat eq uations, Communications on Pure and Applied Mathematics 38 (1985), no. 3, 297–319

work page 1985

[20] [20]

Gu` es, G

O. Gu` es, G. M´ etivier, M. Williams, and K. Zumbrun,Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 151–244. MR2118476

work page 2005

[21] [21]

Holzegel, S

G. Holzegel, S. Klainerman, J. Speck, and W. W.-Y . Wong, Small-data shock formation in solutions to 3D quasilinear w ave equations: an overview, J. Hyperbolic Differ. Equ. 13 (2016), no. 1, 1–105. MR3474069

work page 2016

[22] [22]

John, F ormation of singularities in one-dimensional nonlinear w ave propagation, Comm

F. John, F ormation of singularities in one-dimensional nonlinear w ave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405. MR0369934

work page 1974

[23] [23]

Klainerman and I

S. Klainerman and I. Rodnianski, Improved local well-posedness for quasilinear wave equati ons in dimension three , Duke Math. J. 117 (2003), no. 1, 1–124. MR1962783

work page 2003

[24] [24]

Klainerman, Long time behaviour of solutions to nonlinear wave equation s, Proceedings of the International Congress of Mathematicians, Vol

S. Klainerman, Long time behaviour of solutions to nonlinear wave equation s, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 1984, pp. 1209–12 15. MR804771

work page 1983

[25] [25]

Klainerman and A

S. Klainerman and A. Majda, F ormation of singularities for wave equations including the nonlinear vibrating string , Comm. Pure Appl. Math. 33 (1980), no. 3, 241–263. MR562736

work page 1980

[26] [26]

P . D. Lax, Development of singularities of solutions of nonlinear hyp erbolic partial differential equations , J. Mathematical Phys. 5 (1964), 611–613. MR0165243

work page 1964

[27] [27]

T. P . Liu, Development of singularities in the nonlinear waves for qua silinear hyperbolic partial differential equations , J. Differential Equations 33 (1979), no. 1, 92–111. MR540819

work page 1979

[28] [28]

Luk and J

J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math. 214 (2018), no. 1, 1–169. MR3858399

work page 2018

[29] [29]

Majda, The existence of multidimensional shock fronts , Mem

A. Majda, The existence of multidimensional shock fronts , Mem. Amer. Math. Soc. 43 (1983), no. 281, v+93. MR699241

work page 1983

[30] [30]

Majda, The stability of multidimensional shock fronts , Mem

A. Majda, The stability of multidimensional shock fronts , Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95. MR683422

work page 1983

[31] [31]

Majda, Compressible ﬂuid ﬂow and systems of conservation laws in se veral space variables , Applied Mathematical Sci- ences, vol

A. Majda, Compressible ﬂuid ﬂow and systems of conservation laws in se veral space variables , Applied Mathematical Sci- ences, vol. 53, Springer-V erlag, New Y ork, 1984. MR748308

work page 1984

[32] [32]

Martel, F

Y . Martel, F. Merle, and P . Rapha¨ el,Blow up for the critical generalized Korteweg–de Vries equa tion. I: Dynamics near the soliton, Acta Math. 212 (2014), no. 1, 59–140. MR3179608

work page 2014

[33] [33]

Merle, Asymptotics for L2 minimal blow-up solutions of critical nonlinear Schr¨ odinger equation , Ann

F. Merle, Asymptotics for L2 minimal blow-up solutions of critical nonlinear Schr¨ odinger equation , Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire13 (1996), no. 5, 553–565. MR1409662

work page 1996

[34] [34]

Merle and P

F. Merle and P . Raphael, The blow-up dynamic and upper bound on the blow-up rate for cr itical nonlinear Schr¨ odinger equation, Ann. of Math. (2) 161 (2005), no. 1, 157–222. MR2150386

work page 2005

[35] [35]

Merle, P

F. Merle, P . Rapha¨ el, and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to th e critical Schr¨ odinger map problem, Invent. Math. 193 (2013), no. 2, 249–365. MR3090180

work page 2013

[36] [36]

Merle and H

F. Merle and H. Zaag, Stability of the blow-up proﬁle for equations of the type ut “ ∆u ` | u|p´1u, Duke Math. J. 86 (1997), no. 1, 143–195. MR1427848

work page 1997

[37] [37]

Merle and H

F. Merle and H. Zaag, On the stability of the notion of non-characteristic point a nd blow-up proﬁle for semilinear wave equations, Comm. Math. Phys. 333 (2015), no. 3, 1529–1562. MR3302641

work page 2015

[38] [38]

M´ etivier,Stability of multidimensional shocks , Advances in the theory of shock waves, 2001, pp

G. M´ etivier,Stability of multidimensional shocks , Advances in the theory of shock waves, 2001, pp. 25–103. MR1 842775

work page 2001

[39] [39]

Miao and P

S. Miao and P . Y u, On the formation of shocks for quasilinear wave equations , Invent. Math. 207 (2017), no. 2, 697–831. MR3595936

work page 2017

[40] [40]

Pomeau, M

Y . Pomeau, M. Le Berre, P . Guyenne, and S. Grilli,W ave-breaking and generic singularities of nonlinear hyperbolic equations, Nonlinearity 21 (2008), no. 5, T61–T79. MR2412317

work page 2008

[41] [41]

Riemann, ¨Uber die Fortpﬂanzung ebener Luftwellen von endlicher Schw ingungsweite, Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften in G¨ ottingen8 (1860), 43–66

B. Riemann, ¨Uber die Fortpﬂanzung ebener Luftwellen von endlicher Schw ingungsweite, Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften in G¨ ottingen8 (1860), 43–66

work page

[42] [42]

T. C. Sideris, F ormation of singularities in three-dimensional compress ible ﬂuids , Comm. Math. Phys. 101 (1985), no. 4, 475–485. MR815196

work page 1985

[43] [43]

Speck, Shock formation in small-data solutions to 3D quasilinear w ave equations, Mathematical Surveys and Monographs, vol

J. Speck, Shock formation in small-data solutions to 3D quasilinear w ave equations, Mathematical Surveys and Monographs, vol. 214, American Mathematical Society, Providence, RI, 2 016. MR3561670 39

work page