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

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.

Citation: Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & 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.  Google Scholar

[2]

A. Abbondandolo and P. Majer, Morse homology on Hilbert spaces, Comm. Pure Appl. Math., 54 (2001), 689-760.  doi: 10.1002/cpa.1012.  Google Scholar

[3]

H. BartI. 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.  Google Scholar

[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.  Google Scholar

[5]

R. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194.  doi: 10.1007/BF01206009.  Google Scholar

[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.  Google Scholar

[7]

G. Q. ChenC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

C. LattanzioC. MasciaT. NguyenR. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X.  Google Scholar

[13]

Y. LatushkinA. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. MétivierT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

F. Nazarov, private communication. Google Scholar

[28]

T. NguyenR. 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.  Google Scholar

[29]

D. PeterhofB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocitie, Discrete Contin. Dyn. Syst. A, 20 (2008), 139-158.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

K. Zumbrun, L resolvent estimates for steady Boltzmann's equation, Kinet. Relat. Models, 10 (2017), 1255-1257.  doi: 10.3934/krm.2017048.  Google Scholar

[43]

K. Zumbrun, Conditional stability of unstable viscous shocks, J. Diff. Eq., 247 (2009), 648-671.  doi: 10.1016/j.jde.2009.02.017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. Abbondandolo and P. Majer, Morse homology on Hilbert spaces, Comm. Pure Appl. Math., 54 (2001), 689-760.  doi: 10.1002/cpa.1012.  Google Scholar

[3]

H. BartI. 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.  Google Scholar

[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.  Google Scholar

[5]

R. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194.  doi: 10.1007/BF01206009.  Google Scholar

[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.  Google Scholar

[7]

G. Q. ChenC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

C. LattanzioC. MasciaT. NguyenR. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.  doi: 10.1137/09076026X.  Google Scholar

[13]

Y. LatushkinA. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. MétivierT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

F. Nazarov, private communication. Google Scholar

[28]

T. NguyenR. 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.  Google Scholar

[29]

D. PeterhofB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocitie, Discrete Contin. Dyn. Syst. A, 20 (2008), 139-158.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

K. Zumbrun, L resolvent estimates for steady Boltzmann's equation, Kinet. Relat. Models, 10 (2017), 1255-1257.  doi: 10.3934/krm.2017048.  Google Scholar

[43]

K. Zumbrun, Conditional stability of unstable viscous shocks, J. Diff. Eq., 247 (2009), 648-671.  doi: 10.1016/j.jde.2009.02.017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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