Existence of global classical solutions for the Saint-Venant equations

R. Azib   ; S. Georgiev   ; A. Kheloufi   ; K. Mebarki

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 21-36

Résumé

Nowadays, investigations of the existence of global classical solutions for non linear evolution equations is a topic of active mathematical research. In this article, we are concerned with a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adhémar Jean-Claude Barré de Saint-Venant. Namely, we investigate an initial value problem for the one dimensional Saint-Venant equations. We are especially interested in question of what sufficient conditions the initial data and the topography of the bottom must verify in order that the considered system has global classical solutions. In order to prove our main results we use a new topological approach based on the fixed point abstract theory of the sum of two operators in Banach spaces. This basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics.

Bibliographie

[1] Saint-Venant, A. J. C., “Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et a l'introduction des marées dans leur lit”, Comptes rendus de l'Académie des Sciences de Paris, 73 (1871), 147–154

[2] Baklanovskaya, V. F. and Chechel', I. I., “A Numerical Method for Solving St. Venant's Equations (Chamber Model)”, USSR Computational Mathematics and Mathematical Physics, 16:5 (1976), 125–141 | DOI

[3] Alcrudo, F. and Garcia-Navarro, P., “A High-Resolution Godunov-Type Scheme in Finite Volumes for the 2D Shallow-Water Equations”, International Journal for Numerical Methods in Fluids, 16:6 (1993), 489–505 | DOI

[4] Garcia-Navarro, P., Alcrudo, F. and Savirón, J. M., “1-D Open-Channel Flow Simulation Using TVD-McCormack Scheme”, Journal of Hydraulic Engineering, 118:10 (1992), 1359–1372 | DOI

[5] Lien, F.-S., “A Pressure-Based Unstructured Grid Method for All-Speed Flows”, International Journal for Numerical Methods in Fluids, 33:3 (2000), 355–374 | 3.0.CO;2-X class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[6] Nguyen, D. K., Shi, Y.-E., Wang, S. S. and Nguyen, T. H., “2D Shallow-Water Model Using Unstructured Finite-Volumes Methods”, Journal of Hydraulic Engineering, 132:3 (2006), 258–269 | DOI

[7] Zarmehi, F., Tavakoli, A. and Rahimpour, M., “On Numerical Stabilization in the Solution of Saint-Venant Equations Using the Finite Element Method”, Computers and Mathematics with Applications, 62:4 (2011), 1957–1968 | DOI

[8] Baklanovskaya, V. F., Paltsev, B. V. and Chechel, I. I., “Boundary Value Problems for the St. Venant System of Equations on a Plane”, USSR Computational Mathematics and Mathematical Physics, 19:3 (1979), 157–174 | DOI

[9] Ton, B. A., “Existence and Uniqueness of a Classical Solution of an Initial Boundary Value Problem of the Theory of Shallow Waters”, SIAM Journal on Mathematical Analysis, 12:2 (1981), 229–241 | DOI

[10] Kloeden, P. E., “Global Existence of Classical Solutions in the Dissipative Shallow Water Equations”, SIAM Journal on Mathematical Analysis, 16:2 (1985), 301–315 | DOI

[11] Lai, H. V., Long, L. M. and Trung, M. D., “On Uniqueness of a Classical Solution of the System of Non-Linear 1-D Saint Venant Equations”, Vietnam Journal of Mechanics, NCST of Vietnam, 21:4 (1999), 231–238 | DOI

[12] Muzaev, I. D. and Tuaeva, Zh. D., “Two Methods for Solving an Initial-Boundary Value Problem for a System of Saint-Venant Equations”, Vladikavkaz. Math. J., 1:1 (1999), 43–47 (in Russian)

[13] Li, Y., Pan, R. and Zhu, S., “On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities”, Journal of Mathematical Fluid Mechanics, 19:1 (2017), 151–190 | DOI

[14] Cheng, B. and Tadmor, E., “Long-Time Existence of Smooth Solutions for the Rapidly Rotating Shallow-Water and Euler Equations”, SIAM Journal on Mathematical Analysis, 39:5 (2008), 1668–1685 | DOI

[15] Cheng, B. and Xie, C., “On the Classical Solutions of Two Dimensional Inviscid Rotating Shallow Water System”, Journal of Differential Equations, 250:2 (2011), 690–709 | DOI

[16] Duan, B., Luo, Z. and Zheng, Y., “Local Existence of Classical Solutions to Shallow Water Equations with Cauchy-Data Containing Vacuum”, SIAM Journal on Mathematical Analysis, 44:2 (2012), 541–567 | DOI

[17] Liu, J.-G., Pego, R. L. and Pu, Y., “Well-Posedness and Derivative Blow-Up for a Dispersionless Regularized Shallow Water System”, Nonlinearity, 32:11 (2019), 4346–4376 | DOI

[18] Alekseenko, S. N., Dontsova, M. V. and Pelinovsky, D. E., “Global Solutions to the Shallow Water System with a Method of an Additional Argument”, Applicable Analysis, 96:9 (2017), 1444–1465 | DOI

[19] Yang, Y., “Global Classical Solutions to Two-Dimensional Chemotaxis-Shallow Water System”, Discrete and Continuous Dynamical Systems — B, 26:5 (2021), 2625–2643 | DOI

[20] Licht, C. C., Ha, T. T. and Vu, Q. P., “On Some Linearized Problems of Shallow Water Flows”, Differential and Integral Equations, 22:3–4 (2009), 275–283

[21] Georgiev, S. G. and Zennir, K., “Existence of Solutions for a Class of Nonlinear Impulsive Wave Equations”, Ricerche di Matematica, 71:1 (2022), 211–225 | DOI

[22] Djebali, S. and Mebarki, K., “Fixed Point Index Theory for Perturbation of Expansive Mappings by $k$-Set Contractions”, Topological Methods Nonlinear Analysis, 54:2A (2019), 613–640 | DOI

[23] Polyanin, A. and Manzhirov, A., Handbook of Integral Equations, CRC Press, 1998, 796 pp.