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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 |
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.
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. |
| [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.
|
| [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. |
| [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.
|
| [5] |
F. Colombini, E. Jannelli and S. Spagnolo,
Nonuniqueness in hyperbolic cauchy problems, Ann. of Math., 126 (1987), 495-524.
doi: 10.2307/1971359. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [10] |
F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann, Sci. Ecole Norm. Suph.(4), 22 (1989), 109–125. |
| [11] |
L. Hörmander, Linear Partial Differential Operators, Springer-Velag, Berlin Heidelberg, 1976. |
| [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. |
| [13] |
E. Jannelli,
Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl., 140 (1985), 133-145.
doi: 10.1007/BF01776846. |
| [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., |
| [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. |
| [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. |
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. |
| [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.
|
| [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. |
| [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.
|
| [5] |
F. Colombini, E. Jannelli and S. Spagnolo,
Nonuniqueness in hyperbolic cauchy problems, Ann. of Math., 126 (1987), 495-524.
doi: 10.2307/1971359. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [10] |
F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann, Sci. Ecole Norm. Suph.(4), 22 (1989), 109–125. |
| [11] |
L. Hörmander, Linear Partial Differential Operators, Springer-Velag, Berlin Heidelberg, 1976. |
| [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. |
| [13] |
E. Jannelli,
Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl., 140 (1985), 133-145.
doi: 10.1007/BF01776846. |
| [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., |
| [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. |
| [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. |
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