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doi: 10.3934/dcdsb.2022068
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## Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

 Lebanese American University (LAU), Graduate Studies and Research (GSR), School of Arts and Sciences, Department of Computer Science and Mathematics, Byblos, Lebanon

Received  May 2021 Revised  January 2022 Early access March 2022

In the mathematical theory of water waves, this paper focuses on the hierarchy of higher order asymptotic models. The well-posedness of the medium amplitude extended Green-Naghdi model, as well as higher-ordered Boussinesq-Peregrine and Boussinesq models, is first demonstrated. Introducing a regularization term and various physical topography variations, we show that these models admit unique solutions by a standard energy estimate method in the "hyperbolic" space $H^{s+2}( \mathbb R)^2$, $s>3/2$, but on short/intermediate time scales with respect to amplitude and topography parameters of order $\varepsilon^{-1/4}$, $\max(\sqrt{ \varepsilon}, \beta)^{-1} = \beta^{-1}$ and $\varepsilon^{-1/2}$ respectively. Furthermore, we show that the extended Green-Naghdi system's long-term well-posedness is reachable on time scales of order $\max( \varepsilon, \beta)^{-1}$. The above three specified models, in particular, admit longer time existence of order $\varepsilon^{-1}$, $\max( \varepsilon, \beta)^{-1}$, and $\varepsilon^{-1}$, respectively.

Citation: Bashar Khorbatly. Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022068
##### References:
 [1] S. Alinhac and P. Gérard, Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser, Savoirs Actuels. [Current Scholarship], InterEditions, Paris; Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon, 1991. [2] B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57 (2008), 97-131.  doi: 10.1512/iumj.2008.57.3200. [3] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4. [4] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010. [5] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1. [6] J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108. [7] V. Duchêne and S. Israwi, Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems, Ann. Math. Blaise Pascal, 25 (2018), 21-74.  doi: 10.5802/ambp.372. [8] A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Roy. Soc. London Ser. A, 338 (1974), 43-55.  doi: 10.1098/rspa.1974.0072. [9] A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, Journal of Fluid Mechanics, 78 (1976), 237-246. [10] T. Iguchi, Isobe–kakinuma model for water waves as a higher order shallow water approximation, J. Differential Equations, 265 (2018), 935-962.  doi: 10.1016/j.jde.2018.03.019. [11] T. Iguchi, A mathematical justification of the isobe–kakinuma model for water waves with and without bottom topography, J. Math. Fluid Mech., 20 (2018), 1985-2018.  doi: 10.1007/s00021-018-0398-x. [12] S. Israwi, Derivation and analysis of a new 2d Green—Naghdi system, Nonlinearity, 23 (2010), 2889-2904.  doi: 10.1088/0951-7715/23/11/009. [13] S. Israwi, Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74 (2011), 81-93.  doi: 10.1016/j.na.2010.08.019. [14] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704. [15] B. Khorbatly, A remark on the well-posedness of the classical Green—Naghdi system, Math. Methods Appl. Sci., 44 (2021), 14545-14555.  doi: 10.1002/mma.7724. [16] B. Khorbatly and S. Israwi, Full justification for the extended Green—Naghdi system for an uneven bottom with/without surface tension, Publications of the Research Institute for Mathematical Sciences, (2022). [17] B. Khorbatly, I. Zaiter and S. Isrwai, Derivation and well-posedness of the extended Green—Naghdi equations for flat bottoms with surface tension, J. Math. Phys., 59 (2018), 071501, 20 pp. doi: 10.1063/1.5020601. [18] B. Khorbatly, R. Lteif, S. Israwi and S. Gerbi, Mathematical modeling and numerical analysis for the higher order Boussinesq system, ESAIM Math. Model. Numer. Anal., 56 (2022), 593-615.  doi: 10.1051/m2an/2022015. [19] D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539.  doi: 10.1016/j.jfa.2005.07.003. [20] D. Lannes, The Water Waves Problem, Mathematical analysis and asymptotics. Mathematical Surveys and Monographs, 188. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188. [21] D. Lannes, Modeling shallow water waves, Nonlinearity, 33 (2020), 1-57.  doi: 10.1088/1361-6544/ab6c7c. [22] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21 (2009), 016601. [23] Y. Matsuno, Hamiltonian formulation of the extended Green-Naghdi equations, Phys. D, 301/302 (2015), 1-7.  doi: 10.1016/j.physd.2015.03.001. [24] Y. Matsuno, Hamiltonian structure for two-dimensional extended Green-Naghdi equations, Proc. A., 472 (2016), 20160127, 24 pp. doi: 10.1098/rspa.2016.0127. [25] J.-C. Saut, C. Wang and L. Xu, The Cauchy problem on large time for surface-waves-type Boussinesq systems Ⅱ, SIAM J. Math. Anal., 49 (2017), 2321-2386. [26] J.-C. Saut and L. Xu, Long time existence for a strongly dispersive boussinesq system, SIAM J. Math. Anal., 52 (2020), 2803-2848.  doi: 10.1137/19M1250698. [27] M. Taylor, Partial Differential Equations II, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.

show all references

##### References:
 [1] S. Alinhac and P. Gérard, Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser, Savoirs Actuels. [Current Scholarship], InterEditions, Paris; Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon, 1991. [2] B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57 (2008), 97-131.  doi: 10.1512/iumj.2008.57.3200. [3] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4. [4] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010. [5] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1. [6] J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108. [7] V. Duchêne and S. Israwi, Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems, Ann. Math. Blaise Pascal, 25 (2018), 21-74.  doi: 10.5802/ambp.372. [8] A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Roy. Soc. London Ser. A, 338 (1974), 43-55.  doi: 10.1098/rspa.1974.0072. [9] A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, Journal of Fluid Mechanics, 78 (1976), 237-246. [10] T. Iguchi, Isobe–kakinuma model for water waves as a higher order shallow water approximation, J. Differential Equations, 265 (2018), 935-962.  doi: 10.1016/j.jde.2018.03.019. [11] T. Iguchi, A mathematical justification of the isobe–kakinuma model for water waves with and without bottom topography, J. Math. Fluid Mech., 20 (2018), 1985-2018.  doi: 10.1007/s00021-018-0398-x. [12] S. Israwi, Derivation and analysis of a new 2d Green—Naghdi system, Nonlinearity, 23 (2010), 2889-2904.  doi: 10.1088/0951-7715/23/11/009. [13] S. Israwi, Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74 (2011), 81-93.  doi: 10.1016/j.na.2010.08.019. [14] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704. [15] B. Khorbatly, A remark on the well-posedness of the classical Green—Naghdi system, Math. Methods Appl. Sci., 44 (2021), 14545-14555.  doi: 10.1002/mma.7724. [16] B. Khorbatly and S. Israwi, Full justification for the extended Green—Naghdi system for an uneven bottom with/without surface tension, Publications of the Research Institute for Mathematical Sciences, (2022). [17] B. Khorbatly, I. Zaiter and S. Isrwai, Derivation and well-posedness of the extended Green—Naghdi equations for flat bottoms with surface tension, J. Math. Phys., 59 (2018), 071501, 20 pp. doi: 10.1063/1.5020601. [18] B. Khorbatly, R. Lteif, S. Israwi and S. Gerbi, Mathematical modeling and numerical analysis for the higher order Boussinesq system, ESAIM Math. Model. Numer. Anal., 56 (2022), 593-615.  doi: 10.1051/m2an/2022015. [19] D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539.  doi: 10.1016/j.jfa.2005.07.003. [20] D. Lannes, The Water Waves Problem, Mathematical analysis and asymptotics. Mathematical Surveys and Monographs, 188. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188. [21] D. Lannes, Modeling shallow water waves, Nonlinearity, 33 (2020), 1-57.  doi: 10.1088/1361-6544/ab6c7c. [22] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21 (2009), 016601. [23] Y. Matsuno, Hamiltonian formulation of the extended Green-Naghdi equations, Phys. D, 301/302 (2015), 1-7.  doi: 10.1016/j.physd.2015.03.001. [24] Y. Matsuno, Hamiltonian structure for two-dimensional extended Green-Naghdi equations, Proc. A., 472 (2016), 20160127, 24 pp. doi: 10.1098/rspa.2016.0127. [25] J.-C. Saut, C. Wang and L. Xu, The Cauchy problem on large time for surface-waves-type Boussinesq systems Ⅱ, SIAM J. Math. Anal., 49 (2017), 2321-2386. [26] J.-C. Saut and L. Xu, Long time existence for a strongly dispersive boussinesq system, SIAM J. Math. Anal., 52 (2020), 2803-2848.  doi: 10.1137/19M1250698. [27] M. Taylor, Partial Differential Equations II, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.
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