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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 & 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,, \emph{Netw. Heterog. Media}, 4 (2009), 177. 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,, \emph{Analysis and Simulation of Fluid Dynamics, (2006), 15. 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,, \emph{Mech. Research Comm.}, 38 (2011), 382.

[4]

C.M. Dafermos, Hyperbolic Systems of Conservation Laws,, Springer Verlag, (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, (2011).

[6]

J. F. Gerbeau and B. Perthame, Derivation of viscous saint-venant system for laminar shallow water; numerical validation,, \emph{Disc. Cont. Dyn. Syst.}, 1 (2001), 89. doi: 10.3934/dcdsb.2001.1.89.

[7]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 106 (1987), 169. 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,, \emph{Comm. Math. Phys.}, 281 (2008), 401. 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,, \emph{J. Differential Equations}, 248 (2010), 1926. doi: 10.1016/j.jde.2009.11.029.

[10]

P. L. Lions, Topics in Fluids Mechanics,, Vol. 1 and 2, (1996).

[11]

P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems,, \emph{Comm. Math. Phys.}, 163 (1994), 415.

[12]

P. L Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations,, \emph{J. Amer. Math. Soc.}, 7 (1994), 169. doi: 10.2307/2152725.

[13]

G. Lyng and K. Zumbrun, One-dimensional stability of viscous strong detonation waves,, \emph{Ration. Mech. Anal.}, 173 (2004), 213. doi: 10.1007/s00205-004-0317-6.

[14]

C. Mascia, A dive into shallow water,, \emph{Riv. Mat. Univ. Parma}, 1 (2010), 77.

[15]

C. Mascia and F. Rousset, Asymptotic stability of steady-states for saint-venant equations with real viscosity,, in \emph{Analysis and simulation of fluid dynamics}, (2007), 155. 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,, \emph{Arch. Ration. Mech. Anal.}, 172 (2004), 93. doi: 10.1007/s00205-003-0293-2.

[17]

C. Mascia and K. Zumbrun, Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems,, \emph{Comm. Pure Appl. Math.}, 52 (2004), 841. 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,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 73 (1871), 147.

[19]

W. Wang and C.J. Xu, The Cauchy problem for viscous Shallow Water flows,, \emph{Rev. Mate. Iber.}, 21 (2005), 1. 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,, \emph{Netw. Heterog. Media}, 4 (2009), 177. 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,, \emph{Analysis and Simulation of Fluid Dynamics, (2006), 15. 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,, \emph{Mech. Research Comm.}, 38 (2011), 382.

[4]

C.M. Dafermos, Hyperbolic Systems of Conservation Laws,, Springer Verlag, (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, (2011).

[6]

J. F. Gerbeau and B. Perthame, Derivation of viscous saint-venant system for laminar shallow water; numerical validation,, \emph{Disc. Cont. Dyn. Syst.}, 1 (2001), 89. doi: 10.3934/dcdsb.2001.1.89.

[7]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 106 (1987), 169. 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,, \emph{Comm. Math. Phys.}, 281 (2008), 401. 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,, \emph{J. Differential Equations}, 248 (2010), 1926. doi: 10.1016/j.jde.2009.11.029.

[10]

P. L. Lions, Topics in Fluids Mechanics,, Vol. 1 and 2, (1996).

[11]

P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems,, \emph{Comm. Math. Phys.}, 163 (1994), 415.

[12]

P. L Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations,, \emph{J. Amer. Math. Soc.}, 7 (1994), 169. doi: 10.2307/2152725.

[13]

G. Lyng and K. Zumbrun, One-dimensional stability of viscous strong detonation waves,, \emph{Ration. Mech. Anal.}, 173 (2004), 213. doi: 10.1007/s00205-004-0317-6.

[14]

C. Mascia, A dive into shallow water,, \emph{Riv. Mat. Univ. Parma}, 1 (2010), 77.

[15]

C. Mascia and F. Rousset, Asymptotic stability of steady-states for saint-venant equations with real viscosity,, in \emph{Analysis and simulation of fluid dynamics}, (2007), 155. 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,, \emph{Arch. Ration. Mech. Anal.}, 172 (2004), 93. doi: 10.1007/s00205-003-0293-2.

[17]

C. Mascia and K. Zumbrun, Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems,, \emph{Comm. Pure Appl. Math.}, 52 (2004), 841. 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,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 73 (1871), 147.

[19]

W. Wang and C.J. Xu, The Cauchy problem for viscous Shallow Water flows,, \emph{Rev. Mate. Iber.}, 21 (2005), 1. doi: 10.4171/RMI/412.

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