# American Institute of Mathematical Sciences

• Previous Article
Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms
• CPAA Home
• This Issue
• Next Article
Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy
July  2019, 18(4): 1891-1919. doi: 10.3934/cpaa.2019088

## Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients

 1 School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing, 210023, China 2 School of Mathematics and Physics, Anhui University of Technology, Maanshan 243032, China

* Corresponding author

Received  August 2018 Revised  November 2018 Published  January 2019

Fund Project: The first author and the second author are supported by the NSFC (No.11571177, No.11731007) and the NSF of the Jiangsu Higher Education Institutions of China (17KJA110002)

In this paper, we establish the local well-posedness of low regularity solutions to the general second order strictly hyperbolic equation of divergence form $\partial _t(a_0 \partial _t u)+ \mathop \sum \limits_{j = 1}^n [ \partial _t(a_j \partial _j u)+ \partial _j(a_j \partial _t u)] -\mathop \sum \limits_{j,k = 1}^n \partial _j(a_{jk} \partial _k u)$ $+b_0 \partial _t u+ \partial _t(c_0u)+ \mathop \sum \limits_{j = 1}^n [b_j \partial _ju+ \partial _j(c_ju)] +du = f$ in domain $\Omega = (0, T_0)\times \mathbb R ^n$, where the coefficients $a_0, a_j, a_{jk}\in L^\infty( \Omega )\cap LL(\bar\Omega)$ $(1\le j, k\le n)$, $b_0, c_0, b_j, c_j\in L^\infty( \Omega )\cap C^ \alpha (\bar\Omega)$ $(1\le j\le n)$ for $\alpha \in(\frac{1}{2},1)$, $d\in L^\infty(\Omega)$, $(u(0,x), Xu(0,x))\in (H^{1- \theta +\beta \log}, H^{- \theta +\beta \log})$ with $\theta\in (1- \alpha , \alpha )$, $\beta\in\Bbb R$, and $Xu = a_0 \partial _tu+ \mathop \sum \limits_{j = 1}^n a_j \partial _ju$. Compared with previous references, except a little more general initial data in the space $(H^{1- \theta +\beta \log}, H^{- \theta +\beta \log})$ (only $\beta = 0$ is considered as before), we improve both the lifespan of $u$ up to the precise number $T^*$ and the range of $\theta$ to the left endpoint $1- \alpha$ under some suitable conditions.

Citation: Wenming Hu, Huicheng Yin. Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1891-1919. doi: 10.3934/cpaa.2019088
##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Velag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar [2] J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, (French) [Symbolic calculus and propagation of singularities for non-linear partial differential equations], Ann.Sci. École Norm. Sup., 14 (1981), 209-246. Google Scholar [3] M. Cicognani and F. Colombini, Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem, J. Diff. Eqs., 221 (2006), 143-157. doi: 10.1016/j.jde.2005.06.019. Google Scholar [4] F. Colombini, E. De Giorgi and S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne ${\acute{\rm{q}}}$ue du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 511-559. Google Scholar [5] F. Colombini, E. Jannelli and S. Spagnolo, Nonuniqueness in hyperbolic cauchy problems, Ann. of Math., 126 (1987), 495-524. doi: 10.2307/1971359. Google Scholar [6] F. Colombini and N. Lerner, Operators with non-Lipschitz coefficients, Duke Math. J., 77 (1995), 657-698. doi: 10.1215/S0012-7094-95-07721-7. Google Scholar [7] F. Colombini and G. Métivier, The cauchy problem for wave equations with non lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations, Ann. Sci. Éc. Norm. Supér.(4) , 41 (2008), 177–220. doi: 10.24033/asens.2066. Google Scholar [8] F. Colombini and D. D. Santo, A note on hyperbolic operators with log-zygmund coefficients, J. Math. Sci. Univ. Tokyo, 16 (2009), 95-111. Google Scholar [9] F. Colombini, D.D. Santo, F. Fanelli and G. Métivier, Time-dependent loss of derivatives for hyperbolic operators with non regular coefficients, Comm. Partial Differential Equations, 38 (2013), 1791-1817. doi: 10.1080/03605302.2013.795968. Google Scholar [10] F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann, Sci. Ecole Norm. Suph.(4), 22 (1989), 109–125. Google Scholar [11] L. Hörmander, Linear Partial Differential Operators, Springer-Velag, Berlin Heidelberg, 1976. Google Scholar [12] A. E. Hurd and D. H. Sattinger, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Amer. Math. Soc., 132 (1968), 159-174. doi: 10.2307/1994888. Google Scholar [13] E. Jannelli, Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl., 140 (1985), 133-145. doi: 10.1007/BF01776846. Google Scholar [14] T. Nishitani, Sur les équations hyperboliques á coefficients höldériens en t et de classe de Gevrey en x, Bull. Sci. Math., 107 (1983), 113–138., Google Scholar [15] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, 133, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/133. Google Scholar [16] M. E. Taylor, Tools for PDE : Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000. Google Scholar

show all references

##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Velag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar [2] J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, (French) [Symbolic calculus and propagation of singularities for non-linear partial differential equations], Ann.Sci. École Norm. Sup., 14 (1981), 209-246. Google Scholar [3] M. Cicognani and F. Colombini, Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem, J. Diff. Eqs., 221 (2006), 143-157. doi: 10.1016/j.jde.2005.06.019. Google Scholar [4] F. Colombini, E. De Giorgi and S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne ${\acute{\rm{q}}}$ue du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 511-559. Google Scholar [5] F. Colombini, E. Jannelli and S. Spagnolo, Nonuniqueness in hyperbolic cauchy problems, Ann. of Math., 126 (1987), 495-524. doi: 10.2307/1971359. Google Scholar [6] F. Colombini and N. Lerner, Operators with non-Lipschitz coefficients, Duke Math. J., 77 (1995), 657-698. doi: 10.1215/S0012-7094-95-07721-7. Google Scholar [7] F. Colombini and G. Métivier, The cauchy problem for wave equations with non lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations, Ann. Sci. Éc. Norm. Supér.(4) , 41 (2008), 177–220. doi: 10.24033/asens.2066. Google Scholar [8] F. Colombini and D. D. Santo, A note on hyperbolic operators with log-zygmund coefficients, J. Math. Sci. Univ. Tokyo, 16 (2009), 95-111. Google Scholar [9] F. Colombini, D.D. Santo, F. Fanelli and G. Métivier, Time-dependent loss of derivatives for hyperbolic operators with non regular coefficients, Comm. Partial Differential Equations, 38 (2013), 1791-1817. doi: 10.1080/03605302.2013.795968. Google Scholar [10] F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann, Sci. Ecole Norm. Suph.(4), 22 (1989), 109–125. Google Scholar [11] L. Hörmander, Linear Partial Differential Operators, Springer-Velag, Berlin Heidelberg, 1976. Google Scholar [12] A. E. Hurd and D. H. Sattinger, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Amer. Math. Soc., 132 (1968), 159-174. doi: 10.2307/1994888. Google Scholar [13] E. Jannelli, Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl., 140 (1985), 133-145. doi: 10.1007/BF01776846. Google Scholar [14] T. Nishitani, Sur les équations hyperboliques á coefficients höldériens en t et de classe de Gevrey en x, Bull. Sci. Math., 107 (1983), 113–138., Google Scholar [15] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, 133, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/133. Google Scholar [16] M. E. Taylor, Tools for PDE : Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, American Mathematical Society, Providence, RI, 2000. Google Scholar
 [1] Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321 [2] Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563 [3] Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635 [4] Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669 [5] Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261 [6] Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903 [7] Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081 [8] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010 [9] E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $\mathbb{R}^+$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156 [10] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [11] Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 [12] Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527 [13] A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469 [14] Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195 [15] Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 [16] Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997 [17] Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030 [18] Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763 [19] Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2179-2202. doi: 10.3934/cpaa.2016033 [20] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

2018 Impact Factor: 0.925