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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 |
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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