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On the global well-posedness and scattering of the 3D Klein–Gordon–Zakharov system

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In this paper we are interested in the global well-posedness of the 3D Klein–Gordon–Zakharov equations with small non-compactly supported initial data. We show the uniform boundedness of the energy for the global solution without any compactness assumptions on the initial data. The main novelty of our proof is to apply a modified Alinhac’s ghost weight method together with a newly developed normal-form type estimate to remedy the lack of the space-time scaling vector field; moreover, we give a clear description of the smallness conditions on the initial data.

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Bachelot, A.: Problème de Cauchy global pour des systèmes de Dirac–Klein–Gordon. Ann. Inst. Henri Poincaré 48, 387–422 (1988)

    MathSciNet  Google Scholar 

  4. Cheng, X., Li, D., Xu, J.: Uniform Boundedness of Highest Norm for 2D Quasilinear Wave. Preprint, arXiv: 2104.10019

  5. Cheng, X., Li, D., Xu, J., Zha, D.: Global wellposedness for 2D quasilinear wave without Lorentz. Dynam. Part. Differ. Eq. 19(2), 123–140 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Dendy, R.O.: Plasma Dynamics. Oxford University Press, Oxford (1990)

    Book  Google Scholar 

  8. Dong, S., LeFloch, P., Lei, Z.: The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded. Fundamental Research, (2022)

  9. Dong, S., Ma, Y.: Global Existence and Scattering of the Klein-Gordon-Zakharov System in Two Space Dimensions, arXiv:2111.00244, to appear in Peking Mathematical Journal

  10. Dong, S.: Asymptotic behavior of the solution to the Klein–Gordon–Zakharov model in dimension two. Comm. Math. Phys. 384, 587–607 (2021)

    Article  MathSciNet  Google Scholar 

  11. Dong, S.: Global solution to the Klein–Gordon–Zakharov equations with uniform energy bounds. SIAM J. Math. Anal. 54(1), 595–615 (2022)

    Article  MathSciNet  Google Scholar 

  12. Dong, S., Wyatt, Z.: Stability of a coupled wave-Klein–Gordon system with quadratic nonlinearities. J. Diff. Equ. 269, 7470–7497 (2020)

    Article  MathSciNet  Google Scholar 

  13. Dong, S., Li, K., Yuan, X.: Global solution to the 3D Dirac–Klein–Gordon system with uniform energy bounds. Calc. Var. Part. Diff. Equ. 62(5), 1–42 (2023)

    MathSciNet  Google Scholar 

  14. Duan, S., Ma, Y.: Global solutions of Wave–Klein–Gordon systems in 2+1 dimensional space-time with strong couplings in divergence form. SIAM J. Math. Anal. 54(3), 2691–2726 (2022)

    Article  MathSciNet  Google Scholar 

  15. Georgiev, V.: Global solution of the system of wave and Klein–Gordon equations. Math. Z. 203, 683–698 (1990)

    Article  MathSciNet  Google Scholar 

  16. Georgiev, V.: Decay estimates for the Klein–Gordon equation. Comm. Part. Diff. Equ. 17(7–8), 1111–1139 (1992)

    MathSciNet  Google Scholar 

  17. Guo, B., Yuan, G.: Global smooth solution for the Klein–Gordon–Zakharov equations. J. Math. Phys., 36, (1995)

  18. Guo, Z., Nakanishi, K., Wang, S.: Small energy scattering for the Klein–Gordon–Zakharov system with radial symmetry. Math. Res. Lett. 21, 733–755 (2014)

    Article  MathSciNet  Google Scholar 

  19. Guo, Z., Nakanishi, K., Wang, S.: Global dynamics below the ground state energy for the Klein–Gordon–Zakharov system in the 3D radial case. Comm. Part. Diff. Equ. 39, 1158–1184 (2014)

    Article  MathSciNet  Google Scholar 

  20. Hani, Z., Pusateri, F., Shatah, J.: Scattering for the Zakharov system in 3 dimensions. Comm. Math. Phys. 322, 731–753 (2013)

    Article  MathSciNet  Google Scholar 

  21. Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26. Springer-Verlag, Berlin (1997)

    Google Scholar 

  22. John, F.: Blow-up for quasi-linear wave equations in three space dimensions. Comm. Pure Appl. Math. 34(1), 29–51 (1981)

    Article  MathSciNet  Google Scholar 

  23. Katayama, S.: Global existence for coupled systems of nonlinear wave and Klein–Gordon equations in three space dimensions. Math. Z. 270, 487–513 (2012)

    Article  MathSciNet  Google Scholar 

  24. Klainerman, S.: The null condition and global existence to nonlinear wave equations. Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), 293–326, Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, (1986)

  25. Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein–Gordon equations in four spacetime dimensions. Comm. Pure Appl. Math. 38, 631–641 (1985)

    Article  MathSciNet  Google Scholar 

  26. Klainerman, S., Wang, Q., Yang, S.: Global solution for massive Maxwell–Klein–Gordon equations. Commun. Pure Appl. Math. 73, 63–109 (2020)

    Article  MathSciNet  Google Scholar 

  27. LeFloch, P.G., Ma, Y.: The global nonlinear stability of Minkowski space for self-gravitating massive fields. The wave-Klein–Gordon model. Comm. Math. Phys. 346, 603–665 (2016)

    Article  MathSciNet  Google Scholar 

  28. Li, D.: Uniform estimates for 2D quasilinear wave. Adv. Math. 428, 109157 (2023)

    Article  MathSciNet  Google Scholar 

  29. Li, D., Wu, Y.: The Cauchy problem for the two dimensional Euler-Poisson system. J. Eur. Math. Soc. 16(10), 2211–2266 (2014)

    Article  MathSciNet  Google Scholar 

  30. Masmoudi, N., Nakanishi, K.: Energy convergence for singular limits of Zakharov type systems. Invent. Math. 172, 535–583 (2008)

    Article  MathSciNet  Google Scholar 

  31. Metcalfe, J., Rhoads, T.: Long-time existence for systems of quasilinear wave equations. La Matematica, 1–48, (2023)

  32. Metcalfe, J., Sogge, C.D.: Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods. SIAM J. Math. Anal. 38, 188–209 (2006)

    Article  MathSciNet  Google Scholar 

  33. Metcalfe, J., Stewart, A.: On a system of weakly null semilinear equations. Anal. Math. Phys. 12(5), 125 (2022)

    Article  MathSciNet  Google Scholar 

  34. Ozawa, T., Tsutaya, K., Tsutsumi, Y.: Normal form and global solutions for the Klein–Gordon–Zakharov equations. Ann. Inst. Henri Poincaré 12, 459–503 (1995)

    Article  MathSciNet  Google Scholar 

  35. Ozawa, T., Tsutaya, K., Tsutsumi, Y.: Well-posedness in energy space for the Cauchy problem of the Klein–Gordon–Zakharov equations with different propagation speeds in three space dimensions. Math. Ann. 313, 127–140 (1999)

    Article  MathSciNet  Google Scholar 

  36. Psarelli, M.: Asymptotic behavior of the solutions of Maxwell–Klein–Gordon field equations in 4-dimensional Minkowski space. Comm. Part. Diff. Equ. 24, 223–272 (1999)

    Article  MathSciNet  Google Scholar 

  37. Psarelli, M.: Time decay of Maxwell-Klein-Gordon equations in 4-dimensional Minkowski space. Comm. Part. Diff. Equ. 24, 273–282 (1999)

    Article  MathSciNet  Google Scholar 

  38. Shatah, J.: Normal forms and quadratic nonlinear Klein–Gordon equations. Comm. Pure Appl. Math. 38, 685–696 (1985)

    Article  MathSciNet  Google Scholar 

  39. Shi, Q., Wang, S.: Klein–Gordon–Zakharov system in energy space: blow-up profile and subsonic limit. Math. Methods Appl. Sci. 42, 3211–3221 (2019)

    Article  MathSciNet  Google Scholar 

  40. Sideris, T.C., Tu, S.-Y.: Global existence for systems of nonlinear wave equations in 3D with multiple speeds. SIAM J. Math. Anal. 33, 477–488 (2001)

    Article  MathSciNet  Google Scholar 

  41. Texier, B.: Derivation of the Zakharov equations. Arch. Ration. Mech. Anal. 184, 121–183 (2007)

    Article  MathSciNet  Google Scholar 

  42. Wang, Q.: An intrinsic hyperboloid approach for Einstein Klein–Gordon equations. J. Diff. Geom. 115, 27–109 (2020)

    MathSciNet  Google Scholar 

  43. Wang, C., Yu, X.: Global existence of null-form wave equations on small asymptotically Euclidean manifolds. J. Funct. Anal. 266, 5676–5708 (2014)

    Article  MathSciNet  Google Scholar 

  44. Yang, S.: Pointwise decay for semilinear wave equations in \({\mathbb{R} }^{1+3}\). J. Funct. Anal. 283(2), 109486 (2022)

    Article  Google Scholar 

  45. Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908–914 (1972)

    Google Scholar 

  46. Zha, D.: Global stability of solutions to two-dimension and one-dimension systems of semilinear wave equations. J. Funct. Anal. 282(1), 1092219 (2022)

    Article  MathSciNet  Google Scholar 

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X. Cheng was partially supported by the Shanghai “Super Postdoc" Incentive Plan (No. 2021014), the International Postdoctoral Exchange Fellowship Program (No. YJ20220071) and China Postdoctoral Science Foundation (Grant No. 2022M710796, 2022T150139). J. Xu is supported by the Basic and Applied Basic Research Foundation of Guangzhou.

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Correspondence to Jiao Xu.

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Appendix A: An alternative approach to to estimate \(\Vert \Gamma ^{\beta }n^0\Vert _{L^2}^2\).

Appendix A: An alternative approach to to estimate \(\Vert \Gamma ^{\beta }n^0\Vert _{L^2}^2\).

In this section we give an alternative approach to estimate \(\Vert \Gamma ^{\beta }n^0\Vert _{L^2}^2\) as in Lemma 3.1. It is clear that \(\Box \Gamma ^\beta n^0=0.\) Therefore it is known that we can write \(\Gamma ^\beta n^0\) in the following mild form:

$$\begin{aligned} \Gamma ^\beta n^0(x)= (\cos tD\ v_0)(x)+\left( \frac{\sin tD}{D}v_1\right) (x), \end{aligned}$$

where \(D=|\nabla |\) and \((v_0,v_1)=(\Gamma ^\beta n^0,\partial _t\Gamma ^\beta n^0)|_{t=0}\). Then it follows that \(\Vert \Gamma ^{\beta } n^0\Vert _2\le \Vert \cos tD\ v_0\Vert _2+\Vert \frac{\sin tD}{D}v_1\Vert _2\). One can easily show from the Fourier side that

$$\begin{aligned} \Vert \cos tD\ v_0\Vert _{L^2_x({\mathbb {R}}^3)}\sim \Vert \cos (t|\xi |) \hat{v_0}(\xi )\Vert _{L^2_\xi ({\mathbb {R}}^3)}\lesssim \Vert \hat{v_0}\Vert _{L^2_\xi ({\mathbb {R}}^3)}\lesssim \Vert v_0\Vert _{L^2_x({\mathbb {R}}^3)}. \end{aligned}$$

Similarly by the Hardy’s inequality,

$$\begin{aligned} \left\| \frac{\sin tD}{D}v_1\right\| _{L^2_x({\mathbb {R}}^3)}^2\lesssim \int _{{\mathbb {R}}^3} \frac{(\sin (t|\xi |)^2}{|\xi |^2} |\hat{v_1}|^2\ d\xi \lesssim \int _{{\mathbb {R}}^3} \frac{|\hat{v_1}|^2}{|\xi |^2} \ d\xi \lesssim \Vert \hat{v_1}\Vert ^2_{H^1_\xi ({\mathbb {R}}^3)}\lesssim \Vert \langle x\rangle v_1\Vert ^2_{L^2_x({\mathbb {R}}^3)}. \end{aligned}$$

Remark A.1

Note that compared to the \(X(\partial )\) trick (conformal energy estimates), this propagator estimate needs an additional assumption:

$$\begin{aligned} \Vert v_0\Vert _2+\Vert \langle x\rangle v_1\Vert _2\lesssim \sum _{j=0}^K\Vert \langle x\rangle ^{j+1}\nabla ^j(\nabla n_0,n_1)\Vert _2\le C\varepsilon . \end{aligned}$$

One clearly sees the difference between (3.14) and (A.4).

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Cheng, X., Xu, J. On the global well-posedness and scattering of the 3D Klein–Gordon–Zakharov system. Calc. Var. 63, 17 (2024).

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