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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 & 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

C.M. Dafermos, Hyperbolic Systems of Conservation Laws, Springer Verlag, New York, 1997. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[10]

P. L. Lions, Topics in Fluids Mechanics, Vol. 1 and 2, Oxford Lectures Series in Math. and its Appl., Oxford 1996 and 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. Mascia, A dive into shallow water, Riv. Mat. Univ. Parma, 1 (2010), 77-149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

C.M. Dafermos, Hyperbolic Systems of Conservation Laws, Springer Verlag, New York, 1997. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. L. Lions, Topics in Fluids Mechanics, Vol. 1 and 2, Oxford Lectures Series in Math. and its Appl., Oxford 1996 and 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. Mascia, A dive into shallow water, Riv. Mat. Univ. Parma, 1 (2010), 77-149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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