# American Institute of Mathematical Sciences

• Previous Article
Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge
• CPAA Home
• This Issue
• Next Article
The classification of constant weighted curvature curves in the plane with a log-linear density
July  2014, 13(4): 1653-1667. doi: 10.3934/cpaa.2014.13.1653

## Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval

 1 Universitat Wuerzburg, Campus Hubland Nord, Emil-Fischer-Strasse 30, 97074 Wuerzburg, Germany

Received  September 2013 Revised  January 2014 Published  February 2014

We study the existence and the uniqueness of a {\bf positive connection}, that is a stationary solution connecting the boundary data, for the initial-boundary value problem for the viscous shallow water system \begin{eqnarray} \partial_t u + \partial_x v=0, \partial_t v+\partial_x( \frac{v^2}{u}+ P(u))= \varepsilon\partial_x ( u \partial_x(\frac{v}{u})) \end{eqnarray} in a bounded interval $(-l,l)$ of the real line. We firstly consider the general case where the term of pressure $P(u)$ satisfies \begin{eqnarray} P(0)=0, P(+\infty)=+\infty, P'(u) \quad and \quad P''(u)>0 \ \forall u >0, \end{eqnarray} and then we show properties of the steady state in the relevant case $P(u)=\kappa u^{\gamma}$, $\gamma>1$. The viscous Saint-Venant system, corresponding to $\gamma=2$, fits in the general framework.
Citation: Marta Strani. Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1653-1667. doi: 10.3934/cpaa.2014.13.1653
##### References:
 [1] G. Bastin, J. M. Coron and B. D'Andréa-Novel, On Lyapunov stability of linearized Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187. doi: 10.3934/nhm.2009.4.177. [2] D. Bresch, B. Desjardins B. and G. Métivier, Recent mathematical results and open problems about shallow water equations, Analysis and Simulation of Fluid Dynamics, Series in Advances in Mathematical Fluid Mechanics, Birkhauser Basel, (2006), pp. 15-31. doi: 10.1007/978-3-7643-7742-7_2. [3] S. A. Chin-Bing, P. M. Jordam and A. Warm-Varnas, A note on the viscous, 1D shallow water equation: Traveling wave phenomena, Mech. Research Comm., 38 (2011), 382-387. [4] C.M. Dafermos, Hyperbolic Systems of Conservation Laws, Springer Verlag, New York, 1997. [5] A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of linear hyperbolic systems of balance laws, Preprint of the 18th IFAC World Congress, Milano (Italy) August 28-September 2, 2011. [6] J. F. Gerbeau and B. Perthame, Derivation of viscous saint-venant system for laminar shallow water; numerical validation, Disc. Cont. Dyn. Syst., 1, (2001), 89-102. doi: 10.3934/dcdsb.2001.1.89. [7] S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194. doi: 10.1017/S0308210500018308. [8] H.-L. Li, J. Li and Z. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4. [9] R. Lian, Z. Guo and H.-L. Li, Dynamical behaviors for 1D compressibe Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 248 (2010), 1926-1954. doi: 10.1016/j.jde.2009.11.029. [10] P. L. Lions, Topics in Fluids Mechanics, Vol. 1 and 2, Oxford Lectures Series in Math. and its Appl., Oxford 1996 and 1998. [11] P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys., 163 (1994), 415-431. [12] P. L Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169-191. doi: 10.2307/2152725. [13] G. Lyng and K. Zumbrun, One-dimensional stability of viscous strong detonation waves, Ration. Mech. Anal., 173 (2004), 213-277. doi: 10.1007/s00205-004-0317-6. [14] C. Mascia, A dive into shallow water, Riv. Mat. Univ. Parma, 1 (2010), 77-149. [15] C. Mascia and F. Rousset, Asymptotic stability of steady-states for saint-venant equations with real viscosity, in Analysis and simulation of fluid dynamics, (2007), 155-162, Adv. Math. Fluid Mech., Birkhauser, Basel. doi: 10.1007/978-3-7643-7742-7_9. [16] C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal., 172 (2004), 93-131. doi: 10.1007/s00205-003-0293-2. [17] C. Mascia and K. Zumbrun, Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems, Comm. Pure Appl. Math., 52 (2004), 841-876. doi: 10.1002/cpa.20023. [18] J.C. Barré De Saint-Venant, Théorie du mouvement non permanent des eaux, avec application aux crues des riviéres et á l'introduction des marées dans leur lit, C. R. Acad. Sci. Paris Sér. I Math., 73 (1871), 147-154. [19] W. Wang and C.J. Xu, The Cauchy problem for viscous Shallow Water flows, Rev. Mate. Iber., 21 (2005), 1-24. doi: 10.4171/RMI/412.

show all references

##### References:
 [1] G. Bastin, J. M. Coron and B. D'Andréa-Novel, On Lyapunov stability of linearized Saint-Venant equations for a sloping channel, Netw. Heterog. Media, 4 (2009), 177-187. doi: 10.3934/nhm.2009.4.177. [2] D. Bresch, B. Desjardins B. and G. Métivier, Recent mathematical results and open problems about shallow water equations, Analysis and Simulation of Fluid Dynamics, Series in Advances in Mathematical Fluid Mechanics, Birkhauser Basel, (2006), pp. 15-31. doi: 10.1007/978-3-7643-7742-7_2. [3] S. A. Chin-Bing, P. M. Jordam and A. Warm-Varnas, A note on the viscous, 1D shallow water equation: Traveling wave phenomena, Mech. Research Comm., 38 (2011), 382-387. [4] C.M. Dafermos, Hyperbolic Systems of Conservation Laws, Springer Verlag, New York, 1997. [5] A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of linear hyperbolic systems of balance laws, Preprint of the 18th IFAC World Congress, Milano (Italy) August 28-September 2, 2011. [6] J. F. Gerbeau and B. Perthame, Derivation of viscous saint-venant system for laminar shallow water; numerical validation, Disc. Cont. Dyn. Syst., 1, (2001), 89-102. doi: 10.3934/dcdsb.2001.1.89. [7] S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194. doi: 10.1017/S0308210500018308. [8] H.-L. Li, J. Li and Z. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4. [9] R. Lian, Z. Guo and H.-L. Li, Dynamical behaviors for 1D compressibe Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 248 (2010), 1926-1954. doi: 10.1016/j.jde.2009.11.029. [10] P. L. Lions, Topics in Fluids Mechanics, Vol. 1 and 2, Oxford Lectures Series in Math. and its Appl., Oxford 1996 and 1998. [11] P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys., 163 (1994), 415-431. [12] P. L Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169-191. doi: 10.2307/2152725. [13] G. Lyng and K. Zumbrun, One-dimensional stability of viscous strong detonation waves, Ration. Mech. Anal., 173 (2004), 213-277. doi: 10.1007/s00205-004-0317-6. [14] C. Mascia, A dive into shallow water, Riv. Mat. Univ. Parma, 1 (2010), 77-149. [15] C. Mascia and F. Rousset, Asymptotic stability of steady-states for saint-venant equations with real viscosity, in Analysis and simulation of fluid dynamics, (2007), 155-162, Adv. Math. Fluid Mech., Birkhauser, Basel. doi: 10.1007/978-3-7643-7742-7_9. [16] C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal., 172 (2004), 93-131. doi: 10.1007/s00205-003-0293-2. [17] C. Mascia and K. Zumbrun, Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems, Comm. Pure Appl. Math., 52 (2004), 841-876. doi: 10.1002/cpa.20023. [18] J.C. Barré De Saint-Venant, Théorie du mouvement non permanent des eaux, avec application aux crues des riviéres et á l'introduction des marées dans leur lit, C. R. Acad. Sci. Paris Sér. I Math., 73 (1871), 147-154. [19] W. Wang and C.J. Xu, The Cauchy problem for viscous Shallow Water flows, Rev. Mate. Iber., 21 (2005), 1-24. doi: 10.4171/RMI/412.
 [1] Jean-Frédéric Gerbeau, Benoit Perthame. Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 89-102. doi: 10.3934/dcdsb.2001.1.89 [2] Marie-Odile Bristeau, Jacques Sainte-Marie. Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 733-759. doi: 10.3934/dcdsb.2008.10.733 [3] Hassen Arfaoui, Faker Ben Belgacem, Henda El Fekih, Jean-Pierre Raymond. Boundary stabilizability of the linearized viscous Saint-Venant system. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 491-511. doi: 10.3934/dcdsb.2011.15.491 [4] Emmanuel Audusse, Fayssal Benkhaldoun, Jacques Sainte-Marie, Mohammed Seaid. Multilayer Saint-Venant equations over movable beds. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 917-934. doi: 10.3934/dcdsb.2011.15.917 [5] Georges Bastin, Jean-Michel Coron, Brigitte d'Andréa-Novel. On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks and Heterogeneous Media, 2009, 4 (2) : 177-187. doi: 10.3934/nhm.2009.4.177 [6] Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic and Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031 [7] E. Audusse. A multilayer Saint-Venant model: Derivation and numerical validation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 189-214. doi: 10.3934/dcdsb.2005.5.189 [8] Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227 [9] Majid Bani-Yaghoub, Chunhua Ou, Guangming Yao. Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2509-2535. doi: 10.3934/dcdss.2020195 [10] Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049 [11] Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic and Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883 [12] Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119 [13] Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018 [14] M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827 [15] Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030 [16] Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097 [17] Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198 [18] Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1 [19] Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134 [20] Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065

2020 Impact Factor: 1.916