Space-time L2 estimates, regularity and almost global existence for elastic waves

Kunio Hidano 1  and Dongbing Zha 2
  • 1 Department of Mathematics, Mie University, Faculty of Education, Mie, Japan
  • 2 Department of Applied Mathematics, Donghua University, 201620, Shanghai, P. R. China
Kunio Hidano
  • Department of Mathematics, Faculty of Education, Mie University, 1577 Kurima-machiya-cho, Tsu, Mie, 514-8507, Japan
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Dongbing Zha
  • Corresponding author
  • Department of Applied Mathematics, Donghua University, Shanghai, 201620, P. R. China
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

In this paper, we first establish a kind of weighted space-time L2 estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L2 estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H3×H2.

  • [1]

    R. Agemi, Global existence of nonlinear elastic waves, Invent. Math. 142 (2000), no. 2, 225–250.

    • Crossref
    • Export Citation
  • [2]

    A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd ed., Texts Appl. Math. 4, Springer, New York, 1993.

  • [3]

    Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math. 124 (2014), 1–38.

    • Crossref
    • Export Citation
  • [4]

    K. Hidano, Small solutions to semi-linear wave equations with radial data of critical regularity, Rev. Mat. Iberoam. 25 (2009), no. 2, 693–708.

  • [5]

    K. Hidano, Regularity and lifespan of small solutions to systems of quasi-linear wave equations with multiple speeds. I: Almost global existence, RIMS Kôkyûroku Bessatsu B65 (2017), 37–61.

  • [6]

    K. Hidano, J. Jiang, S. Lee and C. Wang, Weighted fractional chain rule and nonlinear wave equations with minimal regularity, preprint (2018), https://arxiv.org/abs/1605.06748v3.

  • [7]

    K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 (2012), no. 3–4, 267–306.

  • [8]

    K. Hidano, C. Wang and K. Yokoyama, Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations, Math. Ann. 366 (2016), no. 1–2, 667–694.

    • Crossref
    • Export Citation
  • [9]

    K. Hidano and K. Yokoyama, Space-time L 2 L^{2}-estimates and life span of the Klainerman–Machedon radial solutions to some semi-linear wave equations, Differential Integral Equations 19 (2006), no. 9, 961–980.

  • [10]

    T. Hoshiro, On weighted L 2 L^{2} estimates of solutions to wave equations, J. Anal. Math. 72 (1997), 127–140.

    • Crossref
    • Export Citation
  • [11]

    T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal. 63 (1976), no. 3, 273–294.

  • [12]

    J.-C. Jiang, C. Wang and X. Yu, Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal. 11 (2012), no. 5, 1723–1752.

    • Crossref
    • Export Citation
  • [13]

    S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 112, Chapman & Hall/CRC, Boca Raton, 2000.

  • [14]

    F. John, Formation of singularities in elastic waves, Trends and Applications of Pure Mathematics to Mechanics (Palaiseau 1983), Lecture Notes in Phys. 195, Springer, Berlin (1984), 194–210.

  • [15]

    F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 (1988), no. 5, 615–666.

    • Crossref
    • Export Citation
  • [16]

    M. Keel, H. F. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, J. Anal. Math. 87 (2002), 265–279.

    • Crossref
    • Export Citation
  • [17]

    S. Klainerman, On the work and legacy of Fritz John, 1934–1991, Comm. Pure Appl. Math. 51 (1998), 991–1017.

    • Crossref
    • Export Citation
  • [18]

    S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268.

    • Crossref
    • Export Citation
  • [19]

    S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 (1996), 307–321.

    • Crossref
    • Export Citation
  • [20]

    H. Kubo, Lower bounds for the lifespan of solutions to nonlinear wave equations in elasticity, Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math. 301, Birkhäuser, Basel (2012), 187–212.

  • [21]

    H. Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math. 118 (1996), no. 1, 1–16.

    • Crossref
    • Export Citation
  • [22]

    H. Lindblad, Counterexamples to local existence for quasilinear wave equations, Math. Res. Lett. 5 (1998), no. 5, 605–622.

    • Crossref
    • Export Citation
  • [23]

    H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357–426.

    • Crossref
    • Export Citation
  • [24]

    M. Y. Liu and C. B. Wang, Global existence for some 4-D quasilinear wave equations with low regularity, Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 4, 629–640.

    • Crossref
    • Export Citation
  • [25]

    S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal. 219 (2005), no. 1, 1–20.

    • Crossref
    • Export Citation
  • [26]

    J. Metcalfe, Elastic waves in exterior domains. I. Almost global existence, Int. Math. Res. Not. IMRN 2006 (2006), Article ID 69826.

  • [27]

    J. Metcalfe and C. D. Sogge, Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal. 38 (2006), no. 1, 188–209.

    • Crossref
    • Export Citation
  • [28]

    J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z. 256 (2007), no. 3, 521–549.

    • Crossref
    • Export Citation
  • [29]

    J. Metcalfe and B. Thomases, Elastic waves in exterior domains. II. Global existence with a null structure, Int. Math. Res. Not. IMRN 2007 (2007), no. 10, Article ID rnm034.

  • [30]

    E. Y. Ovcharov, Radial Strichartz estimates with application to the 2-D Dirac–Klein–Gordon system, Comm. Partial Differential Equations 37 (2012), no. 10, 1754–1788.

    • Crossref
    • Export Citation
  • [31]

    G. Ponce and T. C. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 18 (1993), no. 1–2, 169–177.

    • Crossref
    • Export Citation
  • [32]

    T. C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent. Math. 123 (1996), no. 2, 323–342.

    • Crossref
    • Export Citation
  • [33]

    T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2) 151 (2000), no. 2, 849–874.

    • Crossref
    • Export Citation
  • [34]

    T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal. 33 (2001), no. 2, 477–488.

    • Crossref
    • Export Citation
  • [35]

    H. F. Smith and D. Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2) 162 (2005), no. 1, 291–366.

    • Crossref
    • Export Citation
  • [36]

    C. D. Sogge, Lectures on Non-Linear Wave Equations, 2nd ed., International Press, Boston, 2008.

  • [37]

    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, And Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993.

  • [38]

    J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. IMRN 2005 (2005), no. 4, 187–231.

    • Crossref
    • Export Citation
  • [39]

    Q. Wang, A geometric approach for sharp local well-posedness of quasilinear wave equations, Ann. PDE 3 (2017), no. 1, Article ID 12.

  • [40]

    D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 (2017), no. 4, 1947–1965.

    • Crossref
    • Export Citation
  • [41]

    Y. Zhou and Z. Lei, Global low regularity solutions of quasi-linear wave equations, Adv. Differential Equations 13 (2008), no. 1–2, 55–104.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

Search