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 and 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, Inc., Mineola, New York, 2006.

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, vol. 190, Academic Press Inc., Boston, MA, 1993.

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third Edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.

[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), 1035-1056. doi: 10.1016/j.jfa.2014.03.021.

[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), 223-235.

[6]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Volume 1, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.

[7]

T. Kato, Quasilinear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Mathematics, 448 (1975), 25-70. doi: 10.1007/BFb0067080.

[8]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Archive for Rational Mechanics and Analysis, 58 (1975), 181-205. doi: 10.1007/BF00280740.

[9]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Communications in Pure and Applied Mathematics, 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[10]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 11, SIAM, Philadelphia, 1973. doi: 10.1137/1.9781611970562.

[11]

P. D. Lax, Hyperbolic Partial Differential Equations, Lecture Notes, 14, Courant Institute of Mathematical Science, American Mathematical Society, Providence, Rhode Island, 2006.

[12]

A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Text Appl. Math., No. 27, Cambridge University Press, Cambridge, 2002.

[14]

J. L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6.

[15]

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

[16]

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

[17]

S. S. Sritharan, An optimal control problem in exterior hydrodynamics, Proceedings of the Royal Society of Edinburgh, 121 (1992), 5-32. doi: 10.1017/S0308210500014128.

[18]

S. S. Sritharan, An introduction to deterministic and stochastic control of viscous flow, in Optimal Control of Viscous Flow (ed. S. S. Sritharan), SIAM, Philadelphia, (1998), 1-42. doi: 10.1137/1.9781611971415.

[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), 1636-1659. doi: 10.1016/j.spa.2006.04.001.

[20]

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

[21]

M. E. Taylor, Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4419-7049-7.

[22]

M. E. Taylor, Tools for PDE, Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, Vol. 81, American Mathematical Society, 2000.

show all references

References:
[1]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Dover Publications, Inc., Mineola, New York, 2006.

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, vol. 190, Academic Press Inc., Boston, MA, 1993.

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third Edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.

[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), 1035-1056. doi: 10.1016/j.jfa.2014.03.021.

[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), 223-235.

[6]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Volume 1, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.

[7]

T. Kato, Quasilinear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Mathematics, 448 (1975), 25-70. doi: 10.1007/BFb0067080.

[8]

T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Archive for Rational Mechanics and Analysis, 58 (1975), 181-205. doi: 10.1007/BF00280740.

[9]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Communications in Pure and Applied Mathematics, 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[10]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 11, SIAM, Philadelphia, 1973. doi: 10.1137/1.9781611970562.

[11]

P. D. Lax, Hyperbolic Partial Differential Equations, Lecture Notes, 14, Courant Institute of Mathematical Science, American Mathematical Society, Providence, Rhode Island, 2006.

[12]

A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Text Appl. Math., No. 27, Cambridge University Press, Cambridge, 2002.

[14]

J. L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6.

[15]

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

[16]

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

[17]

S. S. Sritharan, An optimal control problem in exterior hydrodynamics, Proceedings of the Royal Society of Edinburgh, 121 (1992), 5-32. doi: 10.1017/S0308210500014128.

[18]

S. S. Sritharan, An introduction to deterministic and stochastic control of viscous flow, in Optimal Control of Viscous Flow (ed. S. S. Sritharan), SIAM, Philadelphia, (1998), 1-42. doi: 10.1137/1.9781611971415.

[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), 1636-1659. doi: 10.1016/j.spa.2006.04.001.

[20]

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

[21]

M. E. Taylor, Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4419-7049-7.

[22]

M. E. Taylor, Tools for PDE, Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, Vol. 81, American Mathematical Society, 2000.

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