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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

Wenming Hu 1,2, and  Huicheng Yin 1,,

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.

Keywords: Strictly hyperbolic equation, low regularity, well-posedness, Log Lipschitz, weak solution, loss of derivative.
Mathematics Subject Classification: 35L70, 35L65, 35L67.
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

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