Pith. sign in

REVIEW 2 major objections 3 minor 69 references

Non-uniqueness for the nonlinear dynamical Lam\'e system

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The nonlinear dynamical Lamé system with two distinct wave speeds admits infinitely many weak solutions in $C^{1,\alpha}$ for any $\alpha<1/60$ in three dimensions (and $\alpha<1/30$ in two), all sharing the same small initial data.

desk verdict Solid convex integration construction for the dynamical Lamé system, but the stated Hölder range and the weak-solution definition both need fixing before the theorem as written is supported. read the letter →

arxiv 2502.07385 v1 pith:EYRE5R6J submitted 2025-02-11 math.AP

Shunkai Mao , Peng Qu This is my paper · ORCID
classification math.AP MSC 35A0235D3035L0535L1535L72
keywords Lamésystemnon-uniquenessconvexintegrationweaksolutionsHölderregularitydoublewavespeedslinearlydegeneratehyperbolicCauchyproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the nonlinear dynamical Lamé system, a model of elastic waves with two distinct wave speeds, is non-unique at low Hölder regularity: the same arbitrarily small initial data can produce infinitely many distinct weak solutions in $C^{1,\alpha}$ for $\alpha<1/60$ in three dimensions and $\alpha<1/30$ in two. This is the first non-uniqueness result of this kind for the Lamé system, built with a convex integration scheme. The central mechanism is a new class of building blocks: planar waves combining a longitudinal component of size $O(1)$ with a transverse correction of size $O(\varepsilon)$, which exist only when the two wave speeds genuinely differ, i.e. $\lambda+\mu>0$. If correct, the result shows that low Hölder regularity does not select a unique solution for this quasi-linear hyperbolic system, even though the nonlinearity satisfies the null condition and classical small-data well-posedness holds in higher regularity.

What carries the argument

The central object is the planar-wave building block of Lemma 3.4: a nontrivial solution $w_{A,f}$ of the linearized Lamé-type equation, expressed as $(f+a_{A,2}f^\perp+a_{A,3}\frac{f}{|f|}\times f^\perp)e^{i\xi_{A,f}}$ with $\xi_{A,f}=f\cdot x-((\lambda+2\mu)|f|^2-c_A)^{1/2}t$. The longitudinal component propagates at speed $\sqrt{\lambda+2\mu}$, while the transverse amplitudes $a_{A,2}$, $a_{A,3}$ are fixed by the algebraic system (3.17)–(3.18), whose solution bounds rely on $\lambda+\mu>0$. These building blocks are inserted through a geometric decomposition of the identity (Lemma 2.2) to cancel the Reynolds error, and the perturbation is added in only one direction at a time, six times per iteration, to control the low–high frequency interactions that cannot be eliminated; this six-step pattern is what makes the 3D argument work.

What would settle it

Compute the algebraic system (3.18) in the equal-speed case $\lambda+\mu=0$ for a generic amplitude tensor $A$: if the $2\times 2$ coefficient matrix becomes singular or the unique solution violates the bound $|a_{A,2}|+|a_{A,3}|\lesssim \varepsilon$, then Lemma 3.4 fails and the non-uniqueness result cannot be obtained by this method. Alternatively, run the same construction with $\lambda+\mu=0$ and check whether the perturbation estimates in Proposition 3.7 degrade as $\lambda+\mu\to 0^+$.

Watch

Extended reading notes

Core claim

The paper establishes that for $\mu>0$ and $\lambda+\mu>0$, for every $0<\alpha<1/60$ (in $d=3$) and $0<\alpha<1/30$ (in $d=2$), the Cauchy problem (1.1) with suitably small initial data has infinitely many distinct weak solutions in $C^{1,\alpha}([0,T]\times\mathbb{T}^d)$. The proof constructs a sequence of approximate solutions whose Reynolds error is driven to zero; at each stage the perturbation is a high-frequency superposition of planar waves chosen so that the quadratic nonlinearity cancels the leading error. The novelty is a building block of Lemma 3.4, a planar wave written as $(f+a_{A,2}f^\perp+a_{A,3}\frac{f}{|f|}\times f^\perp)e^{i\xi_{A,f}}$, where the longitudinal wave travels at the $P$-wave speed and the small transverse amplitudes $a_{A,2}, a_{A,3}$ solve the algebraic system (3.18). This system is solvable precisely when $\lambda+\mu>0$, so the double wave speed property is load-bearing rather than a benign assumption. A bifurcation step then produces two solutions with identical initial data but different values in $L^2$, and repeating the argument yields infinitely many distinct solutions.

Load-bearing premise

The construction requires the two elastic wave speeds to be genuinely different (that is, $\lambda+\mu>0$); if they coincide, the algebraic system fixing the transverse amplitudes of the building blocks cannot be solved with the needed bounds, and the entire iteration collapses.

Editorial extensions

If this is right

  • For any Hölder exponent below $1/60$ in three dimensions and below $1/30$ in two dimensions, the Cauchy problem for the nonlinear dynamical Lamé system is not well-posed: the same small initial data has infinitely many weak solutions in $C^{1,\alpha}$.
  • The double wave speed condition $\lambda+\mu>0$ drives the construction: if the two wave speeds coincide, the algebraic system (3.18) for the building block cannot be solved with the required bounds, so the perturbation scheme collapses.
  • The result transplants the convex-integration non-uniqueness phenomenon from fluid equations to a quasi-linear hyperbolic system with a null-condition nonlinearity, showing that the null condition alone does not enforce uniqueness at $C^{1,\alpha}$ regularity.
  • The bifurcation estimate gives a quantitative lower bound on the $L^2$ distance between two solutions, of order $\delta_{q+1}^{1/2}/\lambda_{q+1}$, and the support of their difference can be localized to a prescribed time interval.
  • In the two-dimensional case, the same argument yields the stronger exponent $\alpha<1/30$, and the three-dimensional result is deduced as a corollary of the two-dimensional construction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One might expect the same double-speed mechanism to be adaptable to other two-speed hyperbolic systems, such as full elastodynamics with distinct wave speeds, provided an analogue of the algebraic system (3.18) remains solvable.
  • The exponent $1/60$ is tied to the specific parameter choices, mollification, and geometric decomposition in the iteration; the paper does not claim this threshold is optimal, and a natural open question is whether it can be raised substantially.
  • A separate question, not addressed by the paper, is whether the constructed non-unique solutions survive additional selection criteria such as an energy admissibility condition; the construction only shows weak-solution non-uniqueness in $C^{1,\alpha}$.
  • Since the initial data are required to be small and the admissible size $\varepsilon$ depends on $\lambda+\mu$, a testable consequence is that the non-uniqueness regime shrinks as the two wave speeds approach each other, potentially disappearing in the equal-speed limit $\lambda+\mu=0$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper proposes a convex-integration construction of Hölder-regular weak solutions to the Cauchy problem for the nonlinear dynamical Lamé system (1.1) on T^3 (and T^2), with the same small initial data. The main claim (Theorem 1.3) is that for μ>0 and λ+μ>0, for every 0<α<1/60 there are infinitely many distinct C^{1,α} weak solutions with the same small initial data. The proof follows the De Lellis–Székelyhidi scheme: an induction on approximate solutions (u_q,c_q,R_q), mollification, a six-step perturbation using planar building blocks whose transverse amplitudes solve an algebraic system (Lemma 3.4), estimates on the Reynolds error (Proposition 5.1), and a bifurcation argument (Proposition 2.4). The paper also announces a two-dimensional analogue with exponent 1/30 (Theorem 1.4).

Significance. The novelty is genuine: applying convex integration to a quasi-linear wave system with double wave speeds, exploiting the linear-degeneracy and null-condition structure of the nonlinearity, is new compared with existing Euler/MHD constructions. The paper includes a detailed algebraic building-block lemma, explicit linear-degeneracy computations, and self-contained appendices; the main technical work is not outsourced. If the construction can be repaired as discussed below, it would give the first non-uniqueness result for the nonlinear dynamical Lamé system in a Hölder class. The proof follows the standard convex-integration pattern and contains a large amount of useful technical detail, which is a strength.

major comments (2)
  1. [Proposition 5.1, §6.3] The claimed range 0<α<1/60 is not supported. Proposition 5.1 requires b̄(β)=(1+12β−36γ)/(48β)>1+6β with γ=(1/3)((1+12β)/24−√(β−6β²)/6). Direct arithmetic at β=1/60 gives γ≈0.00986 and b̄(1/60)≈1.056<1.1=1+6/60. The inequality fails numerically for all β≳0.01588≈1/63. Since Theorem 1.3 fixes β∈(α,1/60) and Proposition 2.3 sets b=b̄(β), the induction cannot close for α in (0.01588,1/60); for such α no admissible β exists. At best the proof establishes the construction up to about 1/63, not up to 1/60. This is an internal parameter inconsistency, not merely an endpoint-limiting issue.
  2. [Definition 1.1; §6.3] The limit obtained in Section 6.3 is only shown to lie in C^{1,α′}([0,T]×T³). Definition 1.1, however, defines a weak solution as a function in C([0,T];H²(T³))∩C¹([0,T];H¹(T³)) with ∂_t²u∈L²(0,T;L²(T³)) and initial data in H²×H¹. Hölder C^{1,α} regularity does not imply H² spatial regularity or L² second time derivatives on T³, and no estimate in the paper supplies these norms. The proof verifies only that the equation holds after passing to the limit in the approximate equations; it never checks the regularity clauses of Definition 1.1. Thus Theorem 1.3 as stated is not established. Either Definition 1.1 must be weakened to a distributional/continuous weak-solution notion consistent with the construction, or additional H² estimates must be provided.
minor comments (3)
  1. [Theorem 1.4] Theorem 1.4 is stated for d=2 with exponent 1/30, but no proof is given; the introduction says the three-dimensional result is a corollary of the two-dimensional one, which is also confusing because the 3D theorem is the one proved. Please either prove Theorem 1.4 explicitly or state it as a remark with the necessary changes.
  2. [Introduction] The notation C^{1,1/60−} and C^{1,1/30−} should be replaced by the precise quantifier 'for every α<1/60' (respectively α<1/30) used in Theorems 1.3 and 1.4, to avoid ambiguity about endpoint regularity.
  3. [Proposition 2.3] The condition b̄(β)>1+6β appears only in Proposition 5.1, but the proof of Proposition 2.3 sets b=b̄(β) and relies on it. Since the admissible β interval is exactly what is at stake, this hypothesis should be stated explicitly in Proposition 2.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convex-integration construction is self-contained, with parameters chosen to close estimates rather than fitted to a target.

full rationale

The paper's central claim is a convex-integration construction of infinitely many C^{1,alpha} weak solutions to the nonlinear dynamical Lame system. The derivation chain is internal: starting from an explicit small plane-wave solution with zero Reynolds error, the authors define frequency and amplitude parameters (lambda_q, delta_q, c_q, a, b, beta, gamma) and then prove inductive propositions that close all estimates. No parameter is fitted to a target solution, and no 'prediction' is extracted from data; the non-uniqueness is produced directly by the sign-flipping bifurcation argument in Proposition 2.4, not by invoking a previously established non-uniqueness theorem. The main external inputs are the geometric decomposition lemma from [28], which is stated explicitly with the concrete decomposition (2.2), and standard mollification and oscillatory-integral estimates from [25,30] and [13]; these are independent tools and do not contain the Lame non-uniqueness conclusion. The paper's self-citations [48,52,56] appear only in a survey sentence listing other systems to which convex integration has been applied, and they are not load-bearing in the proof. The only concerns raised by the manuscript are internal parameter-closure issues, such as whether the condition bar b(beta) > 1 + 6 beta really holds for beta near 1/60, and whether the constructed H"older solution satisfies the stated H^2-based weak formulation; both are correctness or estimate-closure questions, not circularity. The derivation does not reduce by definition or by self-citation to its own inputs.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The proof introduces no new physical entities. The free parameters are technical constants chosen to close the convex integration induction; they are not fitted to data. The weak solution regularity axiom is the point where the paper's claim fails, because the constructed C^{1,alpha} limit does not satisfy the H2 requirement.

free parameters (6)
  • epsilon = small, < epsilon_1(lambda,mu)
    Smallness parameter for initial data; chosen to ensure estimates (3.32)-(3.33) and the induction close.
  • beta = 0<beta<1/60 (3D)
    Holder exponent parameter controlling the rate delta_q = lambda_q^{-2 beta}; chosen in the admissible range.
  • b = bar{b}(beta) = (1+12 beta-36 gamma)/(48 beta)
    Base exponent in lambda_q = 2^{6 ceiling(b^q log_2 a)}; chosen to make (5.1) and the induction close.
  • gamma = gamma(beta) < 1/36
    Exponent in the Reynolds error estimates (2.5); chosen to balance parameters.
  • a = a > a*_0 (large)
    Spectral gap parameter making lambda_{q,i+1}/lambda_{q,i} large; chosen sufficiently large.
  • M = M = 120 M0 > 1
    Constant in inductive estimates (2.4)-(2.5); chosen after Proposition 3.7.
assumptions (4)
  • standard math Existence of the geometric decomposition (Lemma 2.2) for symmetric matrices K near Id into projections onto the six directions
    Adapted from De Lellis-Szekelyhidi [28], used to cancel the Reynolds error via (2.2). The paper verifies the specific coefficients for the six chosen frequencies.
  • domain assumption mu>0 and lambda+mu>0
    Needed to solve the building-block system (3.18).
  • domain assumption Small initial data epsilon<epsilon_1(lambda,mu)
    Ensures the amplitude of the transverse waves is small (3.16) and the induction closes.
  • domain assumption The weak solution class requires u in C([0,T];H2) intersect C^1([0,T];H1) with d_tt u in L2
    Definition 1.1. The constructed limit is only C^{1,alpha}, so this axiom is not satisfied.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-uniqueness for the nonlinear dynamical Lam\'e system." pith.science (2026). https://pith.science/paper/EYRE5R6J

@misc{pith2026250207385,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness for the nonlinear dynamical Lam\'e system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EYRE5R6J}},
  note         = {Machine review of arXiv:2502.07385}
}
abstract

We consider the Cauchy problem for the nonlinear dynamical Lam\'e system with double wave speeds in a $d$-dimensional $(d=2,3)$ periodic domain. Moreover, the equations can be transformed into a linearly degenerate hyperbolic system. We could construct infinitely many continuous solutions in $C^{1,\alpha}$ emanating from the same small initial data for $\alpha<\frac{1}{60}$. The proof relies on the convex integration scheme. We construct a new class of building blocks with compression structure by using the double wave speeds characteristic of the equations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

69 extracted references · 44 canonical work pages

  1. [1]

    Al Baba, C

    H. Al Baba, C. Klingenberg, O. Kreml, V . M´ acha, and S. Markfelder. Nonuniqueness of admissible weak solution to the Riemann problem for the full Euler system in two dimensions. SIAM J. Math. Anal. , 52(2):1729–1760, 2020

    work page 2020

  2. [2]

    X. An, H. Chen, and S. Yin. Low regularity ill-posedness f or elastic waves driven by shock formation. Amer . J. Math., 145(4):1111–1181, 2023

    2023

  3. [3]

    S. Alinhac. Blowup of small data solutions for a quasilin ear wave equation in two space dimensions. Ann. of Math. (2) , 149(1):97–127, 1999

    work page 1999

  4. [4]

    S. Alinhac. Blowup of small data solutions for a class of q uasilinear wave equations in two space dimensions. II. Acta Math., 182(1):1–23, 1999

    work page 1999

  5. [5]

    S. Alinhac. Rank 2 singular solutions for quasilinear wa ve equations. Internat. Math. Res. Notices , 2000(18):955–984, 2000

    work page 2000

  6. [6]

    S. Alinhac. The null condition for quasilinear wave equa tions in two space dimensions. I. Invent. Math., 145(3):597–618, 2001

    2001

  7. [7]

    S. Alinhac. The null condition for quasilinear wave equa tions in two space dimensions. II. Amer . J. Math., 123(6):1071–1101, 2001

    work page 2001

  8. [8]

    Beekie, T

    R. Beekie, T. Buckmaster, and V . Vicol. Weak solutions of ideal MHD which do not conserve magnetic helicity. Ann. PDE , 6(1):Paper No. 1, 40pp, 2020

    work page 2020

Show all 69 references
  1. [9]

    M. I. Belishev and I. Lasiecka. The dynamical Lam´ e syste m: regularity of solutions, boundary controllability and b oundary data continu- ation, ESAIM Control Optim. Calc. V ar ., 8:143–167, 2002

    2002

  2. [10]

    L. Bociu. Local and global wellposedness of weak soluti ons for the wave equation with nonlinear boundary and interi or sources of supercritical exponents and damping. Nonlinear Anal., 71(12):e560–e575, 2009

    2009

  3. [11]

    Bˇ rezina, O

    J. Bˇ rezina, O. Kreml, and V . M´ acha. Non-uniqueness of delta shocks and contact discontinuities in the multi-dime nsional model of Chaplygin gas. NoDEA Nonlinear Differential Equations Appl., 28(2):Paper No. 13, 24, 2021

    2021

  4. [12]

    Buckmaster

    T. Buckmaster. Onsager’s conjecture almost everywher e in time. Comm. Math. Phys. , 333(3):1175–1198, 2015

    2015

  5. [13]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi, Jr., and V . Vicol. Onsager’s conjecture for admissible weak solution s. Comm. Pure Appl. Math., 72(2):229–274, 2019

    2019

  6. [14]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, P . Isett, and L. Sz´ ekelyhi di, Jr.. Anomalous dissipation for 1 /5-H¨ older Euler flows. Ann. of Math. (2) , 182(1):127–172, 2015

    2015

  7. [15]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, and L. Sz´ ekelyhidi, Jr. Di ssipative Euler flows with Onsager-critical spatial regula rity. Comm. Pure Appl. Math., 69(9):1613–1670, 2016

    2016

  8. [16]

    Buckmaster and V

    T. Buckmaster and V . Vicol. Nonuniqueness of weak solut ions to the Navier–Stokes equation. Ann. of Math. (2) , 189(1):101–144, 2019

    2019

  9. [17]

    Cheskidov and X

    A. Cheskidov and X. Luo. Sharp nonuniqueness for the Nav ier–Stokes equations. Invent. Math., 229(3):987–1054, 2022

    2022

  10. [18]

    Chiodaroli, C

    E. Chiodaroli, C. De Lellis, and O. Kreml. Global ill-po sedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. , 68(7):1157–1190, 2015

    2015

  11. [19]

    Chiodaroli and O

    E. Chiodaroli and O. Kreml. Non-uniqueness of admissib le weak solutions to the Riemann problem for isentropic Eule r equations. Nonlinearity, 31(4):1441–1460, 2018

    2018

  12. [20]

    Chiodaroli, O

    E. Chiodaroli, O. Kreml, V . M´ acha, and S. Schwarzacher. Non-uniqueness of admissible weak solutions to the compre ssible Euler equa- tions with smooth initial data. Trans. Amer . Math. Soc., 374(4):2269–2295, 2021

    2021

  13. [21]

    Christodoulou

    D. Christodoulou. Global solutions of nonlinear hyper bolic equations for small initial data. Comm. Pure Appl. Math. , 39(2):267–282, 1986. 24

    1986

  14. [22]

    Cie´ slak, B

    T. Cie´ slak, B. Muha, and S. Trifunovi´ c. Global weak solutions in nonlinear 3D thermoelasticity. Calc. V ar . Partial Differential Equations., 63(1):Paper No. 26, 36 pp, 2024

    2024

  15. [23]

    C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics . Springer-V erlag, 2016

    2016

  16. [24]

    Daneri and L

    S. Daneri and L. Sz´ ekelyhidi, Jr. Non-uniqueness and h -principle for H¨ older-continuous weak solutions of the Eu ler equations. Arch. Ration. Mech. Anal., 224(2):471–514, 2017

    2017

  17. [25]

    De Lellis and H

    C. De Lellis and H. Kwon. On nonuniqueness of H¨ older con tinuous globally dissipative Euler flows. Anal. PDE, 15(8):2003–2059, 2022

    2003

  18. [26]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr.. The Euler equatio ns as a di fferential inclusion. Ann. of Math. (2) , 170(3):1417–1436, 2009

    2009

  19. [27]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr.. On admissibility criteria for weak solutions of the Euler equation. Arch. Ration. Mech. Anal. , 195(1):225–260, 2010

    2010

  20. [28]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr.. Dissipative continuous Euler flows. Invent. Math., 193(2):377–407, 2013

    2013

  21. [29]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr.. Dissipative Euler flows and Onsager’s conjecture. J. Eur . Math. Soc., 16(7):1467–1505, 2014

    2014

  22. [30]

    Giri and H

    V . Giri and H. Kwon. On non-uniqueness of continuous ent ropy solutions to the isentropic compressible Euler equati ons. Arch. Ration. Mech. Anal., 245(2):1213–1283, 2022

    2022

  23. [31]

    V . Giri, H. Kwon, and M. Novack. A wavelet-inspired L3-based convex integration framework for the Euler equation s. Ann. PDE , 10(2):Paper No. 19, 271, 2024

    2024

  24. [32]

    Grafakos

    L. Grafakos. Classical Fourier Analysis. Springer, New Y ork, 2014

    2014

  25. [33]

    Guesmia and S

    S. Guesmia and S. Harkat Lam´ e’s system in large size dom ains: existence, asymptotic behaviour and explicit soluti on. J. Math. Anal. Appl., 536(2):Paper No. 128223, 29 pp, 2024

    2024

  26. [34]

    P . Isett. A proof of Onsager’s conjecture. Ann. of Math. (2) , 188(3):871–963, 2018

    2018

  27. [35]

    F. John. Blow-up for quasilinear wave equations in thre e space dimensions. Comm. Pure Appl. Math. , 34(1):29–51, 1981

    1981

  28. [36]

    F. John. Almost global existence of elastic waves of fini te amplitude arising from small initial disturbances. Comm. Pure Appl. Math. , 41(5):615–666, 1988

    1988

  29. [37]

    Klainerman

    S. Klainerman. The Null Condition and Global Existence to Nonlinear W ave Eq uations. Amer. Math. Soc., Providence, RI, 1986

    1986

  30. [38]

    Klingenberg, O

    C. Klingenberg, O. Kreml, and V . M´ acha, and S. Markfeld er. Shocks make the Riemann problem for the full Euler system in multiple space dimensions ill-posed. Nonlinearity, 33(12):6517–6540, 2020

    2020

  31. [39]

    P . D. Lax. Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math. , 10:537–566, 1957

    1957

  32. [40]

    Lei and Y

    Z. Lei and Y . Zhou. Global low regularity solutions of qu asi-linear wave equations. Adv. Differential Equations, 13(1-2):55–104, 2008

    2008

  33. [41]

    Z. Lei, C. T. Sideris, and Y . Zhou. Almost global existen ce for 2-D incompressible isotropic elastodynamics. Trans. Amer . Math. Soc., 367(11):8175–8197, 2015

    2015

  34. [42]

    Z. Lei. Global well-posedness of incompressible elast odynamics in two dimensions. Comm. Pure Appl. Math., 69(11):2072–2106, 2016

    2016

  35. [43]

    Li and X

    T. Li and X. Y u. Dur´ ee de vie des solutions r´ eguli` eres pour les ´ equations des ondes non lin´ eaires.C. R. Acad. Sci. Paris S´ er . I Math., 309(7):469—472, 1989

    1989

  36. [44]

    Li and X

    T. Li and X. Y u. Life-span of classical solutions to full y nonlinear wave equations. Comm. Partial Differential Equations, 16(6-7):909– 940, 1991

    1991

  37. [45]

    Li and Y

    T. Li and Y . Zhou. Life-span of classical solutions to fu lly nonlinear wave equations. II. Nonlinear Anal., 19(9):833–853, 1992

    1992

  38. [46]

    Li and Y

    T. Li and Y . Zhou. Life-span of classical solutions to no nlinear wave equations in two space dimensions. J. Math. Pures Appl, 73(3):223– 249, 1994

    1994

  39. [47]

    Li and Y

    T. Li and Y . Zhou. Nonlinear W ave Equations.. Shanghai Science and Technical Publishers, Shanghai; Spr inger-V erlag, Berlin, 2017

    2017

  40. [48]

    Y . Li, P . Qu, Z. Zeng, and D. Zhang. Sharp non-uniqueness for the 3D hyperdissipative Navier–Stokes equations: abov e the Lions exponent. J. Math. Pures Appl. (9) , 190:Paper No. 103602, 64 pp, 2024

    2024

  41. [49]

    Y . Li, Z. Zeng, and D. Zhang. Non-uniqueness of weak solu tions to 3D magnetohydrodynamic equations. J. Math. Pures Appl. (9) , 165(9):232-285, 2022

    2022

  42. [50]

    Lindblad

    H. Lindblad. On the lifespan of solutions of nonlinear w ave equations with small initial data. Comm. Pure Appl. Math. , 43(4):445–472, 1990

    1990

  43. [51]

    Lindblad

    H. Lindblad. Counterexamples to local existence for qu asilinear wave equations. Math. Res. Lett., 5(5):605–622, 1998

    1998

  44. [52]

    Luo and P

    T. Luo and P . Qu. Non-uniqueness of weak solutions to 2D h ypoviscous Navier–Stokes equations. J. Differential Equations, 269(4):2896– 2919, 2020

    2020

  45. [53]

    Luo and Z

    T. Luo and Z. Xin. H¨ older continuous solutions to the th ree-dimensional Prandtl system. arXiv: 1804.04285, 2018

    2018 arXiv

  46. [54]

    T. F. Ma, J. G. Mesquita and P . N. Seminario-Huertas. Smo oth dynamics of weakly damped Lam´ e systems with delay. SIAM J. Math. Anal., 53(4):3759–3771, 2021

    2021

  47. [55]

    A. Majda. Systems of Conservation Laws in Several Space V ariables. PWN, Warsaw, 1984

    1984

  48. [56]

    Mao and P

    S. Mao and P . Qu. Non-uniqueness for the compressible Eu ler-Maxwell equations. Calc. V ar . Partial Differential Equations, 63(7): Paper No. 186, 84 pp, 2024

    2024

  49. [57]

    Miao and W

    C. Miao and W. Y e. On the weak solutions for the MHD system s with controllable total energy and cross helicity. J. Math. Pures Appl. (9), 181:190–227, 2024

    2024

  50. [58]

    Novack and V

    M. Novack and V . Vicol. An intermittent Onsager theorem . Invent. Math., 233(1):223–323, 2023

    2023

  51. [59]

    Smith and D

    H. Smith and D. Tataru. Sharp local well-posedness resu lts for the nonlinear wave equation. Ann. of Math. (2) , 162(1):291–366, 2005

    2005

  52. [60]

    T. C. Sideris. Formation of singularities in solutions to nonlinear hyperbolic equations. Arch. Rational Mech. Anal., 86(4):369–381, 1984

    1984

  53. [61]

    T. C. Sideris. Formation of singularities in three-dim ensional compressible fluids. Comm. Math. Phys. , 101(4):475–485, 1985

    1985

  54. [62]

    T. C. Sideris. The null condition and global existence o f nonlinear elastic waves. Invent. Math., 123(2):323–342, 1996

    1996

  55. [63]

    T. C. Sideris. Nonresonance and global existence of pre stressed nonlinear elastic waves. Ann. of Math. (2) , 151(2):849–874, 2000

    2000

  56. [64]

    T. C. Sideris and B. Thomases. Global existence for thre e-dimensional incompressible isotropic elastodynamics v ia the incompressible limit. Comm. Pure Appl. Math., 58(6):750–788, 2005. 25

    2005

  57. [65]

    Q. Wang. A geometric approach for sharp local well-pose dness of quasilinear wave equations. Ann. PDE , 3(1):Paper No. 12, 108pp., 2017

    2017

  58. [66]

    Wang, M.M

    R. Wang, M.M. Freitas, B. Feng, and A.J.A. Ramos. Global attractors and synchronization of coupled critical Lam´ e systems with nonlinear damping. J. Differential Equations, 359:476–513, 2023

    2023

  59. [67]

    D. Zha. Global and almost global existence for general q uasilinear wave equations in two space dimensions. J. Math. Pures Appl. (9) , 123(9):270–299, 2019

    2019

  60. [68]

    Zha and K

    D. Zha and K. Hidano. Global solutions to systems of quas ilinear wave equations with low regularity data and applications. J. Math. Pures Appl. (9), 142(9):146–183, 2020

    2020

  61. [69]

    H. Zhang. Local well-posedness for incompressible neo -Hookean elastic equations in almost critical Sobolev spac es. Calc. V ar . Partial Differential Equations, 63(3):Paper No. 66, 20, 2024. School of MathematicalSciences, Fudan University, China. Email address, Shunkai Mao: 2...

    2024

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.