American Institute of Mathematical Sciences

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

New methods for local solvability of quasilinear symmetric hyperbolic systems

Manil T. Mohan 1, and  Sivaguru S. Sritharan 2,

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.
Keywords: Minty-Browder, Quasilinear symmetric hyperbolic equations, optimal control., local monotonicity, commutator estimates.
Mathematics Subject Classification: Primary: 35L60; Secondary: 35L90, 35L45, 49J2.
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, Inc., Mineola, New York, 2006.  Google Scholar

[2]

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

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third Edition, Springer-Verlag, Berlin, 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), 1035-1056. 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), 223-235.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 891-907. 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, No. 11, SIAM, Philadelphia, 1973. doi: 10.1137/1.9781611970562.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  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), 5-32. 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), SIAM, Philadelphia, (1998), 1-42. 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), 1636-1659. doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[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.  Google Scholar

[21]

M. E. Taylor, Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, New York, 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, Vol. 81, American Mathematical Society, 2000.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third Edition, Springer-Verlag, Berlin, 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), 1035-1056. 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), 223-235.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 891-907. 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, No. 11, SIAM, Philadelphia, 1973. doi: 10.1137/1.9781611970562.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  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), 5-32. 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), SIAM, Philadelphia, (1998), 1-42. 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), 1636-1659. doi: 10.1016/j.spa.2006.04.001.  Google Scholar

[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.  Google Scholar

[21]

M. E. Taylor, Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, New York, 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, Vol. 81, American Mathematical Society, 2000.  Google Scholar

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