Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks

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February 2019, 12(1): 1-36. doi: 10.3934/krm.2019001

Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks

Alin Pogan ^1, and Kevin Zumbrun ^2,,

1.	Miami University, Department of Mathematics, 301 S. Patterson Ave., Oxford, OH 45056, USA
2.	Indiana University, Department of Mathematics, 831 E. Third St., Bloomington, IN 47405, USA

^* Corresponding author: Kevin Zumbrun

Received May 2017 Published July 2018

Fund Project: Research of A. P. was partially supported under the Summer Research Grant program, Miami University and by the Simons Foundation. Research of K.Z. was partially supported under NSF grant no. DMS-0300487.

We construct stable manifolds for a class of singular evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.

Keywords: Boltzmann equation, evolution equation, invariant manifold.

Mathematics Subject Classification: Primary: 47J06; Secondary: 47J35, 76P05.

Citation: Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic and Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001

References:

[1]	A. Abbondandolo and P. Majer, Ordinary differential operators in Hilbert spaces and Fredholm pairs, Math. Z., 243 (2003), 525-562. doi: 10.1007/s00209-002-0473-z.
[2]	A. Abbondandolo and P. Majer, Morse homology on Hilbert spaces, Comm. Pure Appl. Math., 54 (2001), 689-760. doi: 10.1002/cpa.1012.
[3]	H. Bart, I. Gohberg and M. A. Kaashoek, Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators, J. Funct. Anal., 68 (1986), 1-42. doi: 10.1016/0022-1236(86)90055-8.
[4]	G. Boillat and T. Ruggeri, On the shock structure problem for hyperbolic system of balance laws and convex entropy, Continuum Mechanics and Thermodynamics, 10 (1998), 285-292. doi: 10.1007/s001610050094.
[5]	R. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. doi: 10.1007/BF01206009.
[6]	C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988. ⅹⅱ+455 pp. ISBN: 0-387-96637-4. doi: 10.1007/978-1-4612-1039-9.
[7]	G. Q. Chen, C. David Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602.
[8]	A. Dressler and W.-A. Yong, Existence of traveling-wave solutions for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 182 (2006), 49-75. doi: 10.1007/s00205-006-0430-9.
[9]	R. A. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), 797-855. doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1.
[10]	H. Grad, Asymptotic theory of the Boltzmann equation. Ⅱ, 1963 Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Academic Press, New York, 1 (1963), 26-59.
[11]	J. Härterich, Viscous profiles for traveling waves of scalar balance laws: The canard case, Methods and Applications of Analysis, 10 (2003), 97-117. doi: 10.4310/MAA.2003.v10.n1.a6.
[12]	C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206. doi: 10.1137/09076026X.
[13]	Y. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414.
[14]	Y. Latushkin and A. Pogan, The dichotomy theorem for evolution bi-families, J. Diff. Eq., 245 (2008), 2267-2306. doi: 10.1016/j.jde.2008.01.023.
[15]	Y. Latushkin and A. Pogan, The infinite dimensional evans function, J. Funct Anal., 268 (2015), 1509-1586. doi: 10.1016/j.jfa.2014.11.020.
[16]	T. P. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.
[17]	T. P. Liu and S. H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997. doi: 10.1007/s00205-013-0640-x.
[18]	J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dyn. Diff. Eq., 11 (1999), 1-47. doi: 10.1023/A:1021889401235.
[19]	C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J., 51 (2002), 773-904. doi: 10.1512/iumj.2002.51.2212.
[20]	C. Mascia and K. Zumbrun, Spectral stability of weak relaxation shock profiles, Comm. Part. Diff. Eq., 34 (2009), 119-136. doi: 10.1080/03605300802553971.
[21]	C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., 37 (2005), 889-913. doi: 10.1137/S0036141004435844.
[22]	C. Mascia and K. Zumbrun, Pointwise Green's function bounds for shock profiles with degenerate viscosity, Arch. Ration. Mech. Anal., 169 (2003), 177-263. doi: 10.1007/s00205-003-0258-5.
[23]	C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Rat. Mech. Anal., 172 (2004), 93-131. doi: 10.1007/s00205-003-0293-2.
[24]	G. Métivier, T. Texier and K. Zumbrun, Existence of quasilinear relaxation shock profiles in systems with characteristic velocities, Ann. Fac. Sci. Toulouse Math., 21 (2012), 1-23. doi: 10.5802/afst.1327.
[25]	G. Métivier and K. Zumbrun, Existence of semilinear relaxation shocks, J. Math. Pures Appl., 92 (2009), 209-231. doi: 10.1016/j.matpur.2009.05.002.
[26]	G. Métivier and K. Zumbrun, Existence and sharp localization in velocity of small-amplitude Boltzmann shocks, Kinet. Relat. Models, 2 (2009), 667-705. doi: 10.3934/krm.2009.2.667.
[27]	F. Nazarov, private communication.
[28]	T. Nguyen, R. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453. doi: 10.1016/j.physd.2010.01.011.
[29]	D. Peterhof, B. Sandstede and A. Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Diff. Eq., 140 (1997), 266-308. doi: 10.1006/jdeq.1997.3303.
[30]	A. Pogan and A. Scheel, Instability of spikes in the presence of conservation laws, Z. Angew. Math. Phys., 61 (2010), 979-998. doi: 10.1007/s00033-010-0058-3.
[31]	A. Pogan and A. Scheel, Layers in the presence of conservation laws, J. Dyn. Diff. Eq., 24 (2012), 249-287. doi: 10.1007/s10884-012-9248-3.
[32]	A. Pogan and K. Zumbrun, Center manifolds of degenerate evolution equations and existence of small-amplitude kinetic shocks, J. Diff Eq., 264 (2018), 6752-6808. doi: 10.1016/j.jde.2018.01.049.
[33]	J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc., 27 (1995), 1-33. doi: 10.1112/blms/27.1.1.
[34]	B. Sandstede, Stability of traveling waves, in: Handbook of Dynamical Systems, vol. 2, NorthHolland, Amsterdam, 2002, 983-1055. doi: 10.1016/S1874-575X(02)80039-X.
[35]	B. Sandstede and A. Scheel, On the structure of spectra of modulated traveling waves, Math. Nachr., 232 (2001), 39-93. doi: 10.1002/1522-2616(200112)232:1<39::AID-MANA39>3.0.CO;2-5.
[36]	B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocitie, Discrete Contin. Dyn. Syst. A, 20 (2008), 139-158.
[37]	B. Texier and K. Zumbrun, Nash-Moser iteration and singular perturbations, Ann. Inst. H. Poincare Anal. Non Lineaire, 28 (2011), 499-527. doi: 10.1016/j.anihpc.2011.05.001.
[38]	K. Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 47 (2001), 307-516.
[39]	K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, With an Appendix by Helge Kristian Jenssen and Gregory Lyng, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, 3 (2004), 311-533.
[40]	K. Zumbrun, Planar stability criteria for viscous shock waves of systems with real viscosity, Hyperbolic Systems of Balance Laws, 229-326, Lecture Notes in Math., 1911, Springer, Berlin, 2007. doi: 10.1007/978-3-540-72187-1_4.
[41]	K. Zumbrun, Stability and dynamics of viscous shock waves, Nonlinear Conservation Laws and Applications, 123-167, IMA Vol. Math. Appl., 153, Springer, New York, 2011. doi: 10.1007/978-1-4419-9554-4_5.
[42]	K. Zumbrun, L^∞ resolvent estimates for steady Boltzmann's equation, Kinet. Relat. Models, 10 (2017), 1255-1257. doi: 10.3934/krm.2017048.
[43]	K. Zumbrun, Conditional stability of unstable viscous shocks, J. Diff. Eq., 247 (2009), 648-671. doi: 10.1016/j.jde.2009.02.017.
[44]	K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871; Errata, Indiana Univ. Math. J., 51 (2002), 1017-1021 doi: 10.1512/iumj.2002.51.2410.
[45]	K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., 48 (1999), 937-992. doi: 10.1512/iumj.1999.48.1765.

show all references

References:

[1]	A. Abbondandolo and P. Majer, Ordinary differential operators in Hilbert spaces and Fredholm pairs, Math. Z., 243 (2003), 525-562. doi: 10.1007/s00209-002-0473-z.
[2]	A. Abbondandolo and P. Majer, Morse homology on Hilbert spaces, Comm. Pure Appl. Math., 54 (2001), 689-760. doi: 10.1002/cpa.1012.
[3]	H. Bart, I. Gohberg and M. A. Kaashoek, Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators, J. Funct. Anal., 68 (1986), 1-42. doi: 10.1016/0022-1236(86)90055-8.
[4]	G. Boillat and T. Ruggeri, On the shock structure problem for hyperbolic system of balance laws and convex entropy, Continuum Mechanics and Thermodynamics, 10 (1998), 285-292. doi: 10.1007/s001610050094.
[5]	R. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. doi: 10.1007/BF01206009.
[6]	C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988. ⅹⅱ+455 pp. ISBN: 0-387-96637-4. doi: 10.1007/978-1-4612-1039-9.
[7]	G. Q. Chen, C. David Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602.
[8]	A. Dressler and W.-A. Yong, Existence of traveling-wave solutions for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 182 (2006), 49-75. doi: 10.1007/s00205-006-0430-9.
[9]	R. A. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), 797-855. doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1.
[10]	H. Grad, Asymptotic theory of the Boltzmann equation. Ⅱ, 1963 Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Academic Press, New York, 1 (1963), 26-59.
[11]	J. Härterich, Viscous profiles for traveling waves of scalar balance laws: The canard case, Methods and Applications of Analysis, 10 (2003), 97-117. doi: 10.4310/MAA.2003.v10.n1.a6.
[12]	C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206. doi: 10.1137/09076026X.
[13]	Y. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414.
[14]	Y. Latushkin and A. Pogan, The dichotomy theorem for evolution bi-families, J. Diff. Eq., 245 (2008), 2267-2306. doi: 10.1016/j.jde.2008.01.023.
[15]	Y. Latushkin and A. Pogan, The infinite dimensional evans function, J. Funct Anal., 268 (2015), 1509-1586. doi: 10.1016/j.jfa.2014.11.020.
[16]	T. P. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.
[17]	T. P. Liu and S. H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997. doi: 10.1007/s00205-013-0640-x.
[18]	J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dyn. Diff. Eq., 11 (1999), 1-47. doi: 10.1023/A:1021889401235.
[19]	C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J., 51 (2002), 773-904. doi: 10.1512/iumj.2002.51.2212.
[20]	C. Mascia and K. Zumbrun, Spectral stability of weak relaxation shock profiles, Comm. Part. Diff. Eq., 34 (2009), 119-136. doi: 10.1080/03605300802553971.
[21]	C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., 37 (2005), 889-913. doi: 10.1137/S0036141004435844.
[22]	C. Mascia and K. Zumbrun, Pointwise Green's function bounds for shock profiles with degenerate viscosity, Arch. Ration. Mech. Anal., 169 (2003), 177-263. doi: 10.1007/s00205-003-0258-5.
[23]	C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Rat. Mech. Anal., 172 (2004), 93-131. doi: 10.1007/s00205-003-0293-2.
[24]	G. Métivier, T. Texier and K. Zumbrun, Existence of quasilinear relaxation shock profiles in systems with characteristic velocities, Ann. Fac. Sci. Toulouse Math., 21 (2012), 1-23. doi: 10.5802/afst.1327.
[25]	G. Métivier and K. Zumbrun, Existence of semilinear relaxation shocks, J. Math. Pures Appl., 92 (2009), 209-231. doi: 10.1016/j.matpur.2009.05.002.
[26]	G. Métivier and K. Zumbrun, Existence and sharp localization in velocity of small-amplitude Boltzmann shocks, Kinet. Relat. Models, 2 (2009), 667-705. doi: 10.3934/krm.2009.2.667.
[27]	F. Nazarov, private communication.
[28]	T. Nguyen, R. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453. doi: 10.1016/j.physd.2010.01.011.
[29]	D. Peterhof, B. Sandstede and A. Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Diff. Eq., 140 (1997), 266-308. doi: 10.1006/jdeq.1997.3303.
[30]	A. Pogan and A. Scheel, Instability of spikes in the presence of conservation laws, Z. Angew. Math. Phys., 61 (2010), 979-998. doi: 10.1007/s00033-010-0058-3.
[31]	A. Pogan and A. Scheel, Layers in the presence of conservation laws, J. Dyn. Diff. Eq., 24 (2012), 249-287. doi: 10.1007/s10884-012-9248-3.
[32]	A. Pogan and K. Zumbrun, Center manifolds of degenerate evolution equations and existence of small-amplitude kinetic shocks, J. Diff Eq., 264 (2018), 6752-6808. doi: 10.1016/j.jde.2018.01.049.
[33]	J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc., 27 (1995), 1-33. doi: 10.1112/blms/27.1.1.
[34]	B. Sandstede, Stability of traveling waves, in: Handbook of Dynamical Systems, vol. 2, NorthHolland, Amsterdam, 2002, 983-1055. doi: 10.1016/S1874-575X(02)80039-X.
[35]	B. Sandstede and A. Scheel, On the structure of spectra of modulated traveling waves, Math. Nachr., 232 (2001), 39-93. doi: 10.1002/1522-2616(200112)232:1<39::AID-MANA39>3.0.CO;2-5.
[36]	B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocitie, Discrete Contin. Dyn. Syst. A, 20 (2008), 139-158.
[37]	B. Texier and K. Zumbrun, Nash-Moser iteration and singular perturbations, Ann. Inst. H. Poincare Anal. Non Lineaire, 28 (2011), 499-527. doi: 10.1016/j.anihpc.2011.05.001.
[38]	K. Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 47 (2001), 307-516.
[39]	K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, With an Appendix by Helge Kristian Jenssen and Gregory Lyng, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, 3 (2004), 311-533.
[40]	K. Zumbrun, Planar stability criteria for viscous shock waves of systems with real viscosity, Hyperbolic Systems of Balance Laws, 229-326, Lecture Notes in Math., 1911, Springer, Berlin, 2007. doi: 10.1007/978-3-540-72187-1_4.
[41]	K. Zumbrun, Stability and dynamics of viscous shock waves, Nonlinear Conservation Laws and Applications, 123-167, IMA Vol. Math. Appl., 153, Springer, New York, 2011. doi: 10.1007/978-1-4419-9554-4_5.
[42]	K. Zumbrun, L^∞ resolvent estimates for steady Boltzmann's equation, Kinet. Relat. Models, 10 (2017), 1255-1257. doi: 10.3934/krm.2017048.
[43]	K. Zumbrun, Conditional stability of unstable viscous shocks, J. Diff. Eq., 247 (2009), 648-671. doi: 10.1016/j.jde.2009.02.017.
[44]	K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871; Errata, Indiana Univ. Math. J., 51 (2002), 1017-1021 doi: 10.1512/iumj.2002.51.2410.
[45]	K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., 48 (1999), 937-992. doi: 10.1512/iumj.1999.48.1765.

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