June  2016, 5(2): 273-302. doi: 10.3934/eect.2016005

New methods for local solvability of quasilinear symmetric hyperbolic systems

1. 

Department of Mathematics and Statistics, Air Force Institute of Technology, 2950 Hobson Way, Wright Patterson Air Force Base, OH 45433, United States

2. 

Office of the Provost & Vice Chancellor, Air Force Institute of Technology, 2950 Hobson Way, Wright Patterson Air Force Base, OH 45433, United States

Received  January 2016 Revised  May 2016 Published  June 2016

In this work we establish the local solvability of quasilinearsymmetric hyperbolic system using local monotonicity method andfrequency truncation method. The existence of an optimal control isalso proved as an application of these methods.
Citation: Manil T. Mohan, Sivaguru S. Sritharan. New methods for local solvability of quasilinear symmetric hyperbolic systems. Evolution Equations & Control Theory, 2016, 5 (2) : 273-302. doi: 10.3934/eect.2016005
References:
[1]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Dover Publications, (2006).   Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems,, Mathematics in Science and Engineering, (1993).   Google Scholar

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Third Edition, (2010).  doi: 10.1007/978-3-642-04048-1.  Google Scholar

[4]

C. L. Fefferman, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models,, Journal of Functional Analysis, 267 (2014), 1035.  doi: 10.1016/j.jfa.2014.03.021.  Google Scholar

[5]

B. P. W. Fernando, S. S. Sritharan and M. Xu, A simple proof of global solvability of 2-D Navier-Stokes equations in unbounded domains,, Differential and Integral Equations, 23 (2010), 223.   Google Scholar

[6]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Volume 1, (1972).  doi: 10.1007/978-3-642-65161-8.  Google Scholar

[7]

T. Kato, Quasilinear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.  doi: 10.1007/BFb0067080.  Google Scholar

[8]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems,, Archive for Rational Mechanics and Analysis, 58 (1975), 181.  doi: 10.1007/BF00280740.  Google Scholar

[9]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Communications in Pure and Applied Mathematics, 41 (1988), 891.  doi: 10.1002/cpa.3160410704.  Google Scholar

[10]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1973).  doi: 10.1137/1.9781611970562.  Google Scholar

[11]

P. D. Lax, Hyperbolic Partial Differential Equations,, Lecture Notes, 14 (2006).   Google Scholar

[12]

A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, 53 (1984).  doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Text Appl. Math., 27 (2002).   Google Scholar

[14]

J. L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation,, Applied Mathematics and Optimization, 46 (2002), 31.  doi: 10.1007/s00245-002-0734-6.  Google Scholar

[15]

U. Manna, M. T. Mohan and S. S. Sritharan, Stochastic non-resistive magnetohydrodynamic system with Lévy noise,, Submitted for Journal Publication., ().   Google Scholar

[16]

M. T. Mohan and S. S. Sritharan, Stochastic Euler equations of fluid dynamics with Lévy noise,, Submitted for Journal Publication., ().   Google Scholar

[17]

S. S. Sritharan, An optimal control problem in exterior hydrodynamics,, Proceedings of the Royal Society of Edinburgh, 121 (1992), 5.  doi: 10.1017/S0308210500014128.  Google Scholar

[18]

S. S. Sritharan, An introduction to deterministic and stochastic control of viscous flow,, in Optimal Control of Viscous Flow (ed. S. S. Sritharan), (1998), 1.  doi: 10.1137/1.9781611971415.  Google Scholar

[19]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise,, Stochastic Processes and their Applications, 116 (2006), 1636.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[20]

M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE,, Springer Science & Business Media LLC, (1991).  doi: 10.1007/978-1-4612-0431-2.  Google Scholar

[21]

M. E. Taylor, Partial Differential Equations III, Nonlinear Equations,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4419-7049-7.  Google Scholar

[22]

M. E. Taylor, Tools for PDE, Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,, Mathematical Surveys and Monographs, 81 (2000).   Google Scholar

show all references

References:
[1]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Dover Publications, (2006).   Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems,, Mathematics in Science and Engineering, (1993).   Google Scholar

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Third Edition, (2010).  doi: 10.1007/978-3-642-04048-1.  Google Scholar

[4]

C. L. Fefferman, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models,, Journal of Functional Analysis, 267 (2014), 1035.  doi: 10.1016/j.jfa.2014.03.021.  Google Scholar

[5]

B. P. W. Fernando, S. S. Sritharan and M. Xu, A simple proof of global solvability of 2-D Navier-Stokes equations in unbounded domains,, Differential and Integral Equations, 23 (2010), 223.   Google Scholar

[6]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Volume 1, (1972).  doi: 10.1007/978-3-642-65161-8.  Google Scholar

[7]

T. Kato, Quasilinear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.  doi: 10.1007/BFb0067080.  Google Scholar

[8]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems,, Archive for Rational Mechanics and Analysis, 58 (1975), 181.  doi: 10.1007/BF00280740.  Google Scholar

[9]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Communications in Pure and Applied Mathematics, 41 (1988), 891.  doi: 10.1002/cpa.3160410704.  Google Scholar

[10]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1973).  doi: 10.1137/1.9781611970562.  Google Scholar

[11]

P. D. Lax, Hyperbolic Partial Differential Equations,, Lecture Notes, 14 (2006).   Google Scholar

[12]

A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, 53 (1984).  doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge Text Appl. Math., 27 (2002).   Google Scholar

[14]

J. L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation,, Applied Mathematics and Optimization, 46 (2002), 31.  doi: 10.1007/s00245-002-0734-6.  Google Scholar

[15]

U. Manna, M. T. Mohan and S. S. Sritharan, Stochastic non-resistive magnetohydrodynamic system with Lévy noise,, Submitted for Journal Publication., ().   Google Scholar

[16]

M. T. Mohan and S. S. Sritharan, Stochastic Euler equations of fluid dynamics with Lévy noise,, Submitted for Journal Publication., ().   Google Scholar

[17]

S. S. Sritharan, An optimal control problem in exterior hydrodynamics,, Proceedings of the Royal Society of Edinburgh, 121 (1992), 5.  doi: 10.1017/S0308210500014128.  Google Scholar

[18]

S. S. Sritharan, An introduction to deterministic and stochastic control of viscous flow,, in Optimal Control of Viscous Flow (ed. S. S. Sritharan), (1998), 1.  doi: 10.1137/1.9781611971415.  Google Scholar

[19]

S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise,, Stochastic Processes and their Applications, 116 (2006), 1636.  doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[20]

M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE,, Springer Science & Business Media LLC, (1991).  doi: 10.1007/978-1-4612-0431-2.  Google Scholar

[21]

M. E. Taylor, Partial Differential Equations III, Nonlinear Equations,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4419-7049-7.  Google Scholar

[22]

M. E. Taylor, Tools for PDE, Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials,, Mathematical Surveys and Monographs, 81 (2000).   Google Scholar

[1]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061

[2]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[3]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

[4]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[5]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[6]

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

[7]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354

[8]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322

[9]

Julian Koellermeier, Giovanni Samaey. Projective integration schemes for hyperbolic moment equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021008

[10]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[11]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[12]

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

[13]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[14]

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

[15]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

[16]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[17]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[18]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[19]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[20]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]