• 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 & 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.  Google Scholar

[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.  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,, \emph{Mech. Research Comm.}, 38 (2011), 382.   Google Scholar

[4]

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

[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.  Google Scholar

[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.  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,, \emph{Comm. Math. Phys.}, 281 (2008), 401.  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,, \emph{J. Differential Equations}, 248 (2010), 1926.  doi: 10.1016/j.jde.2009.11.029.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 73 (1871), 147.   Google Scholar

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

[4]

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

[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.  Google Scholar

[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.  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,, \emph{Comm. Math. Phys.}, 281 (2008), 401.  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,, \emph{J. Differential Equations}, 248 (2010), 1926.  doi: 10.1016/j.jde.2009.11.029.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 73 (1871), 147.   Google Scholar

[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.  Google Scholar

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[2]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[3]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[4]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[5]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[6]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[7]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[8]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[9]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[10]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[11]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[12]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[13]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[14]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[15]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[16]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[17]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[18]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[19]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[20]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]