December  2019, 24(12): 6349-6365. doi: 10.3934/dcdsb.2019142

On the $ L^p $ regularity of solutions to the generalized Hunter-Saxton system

1. 

Department of Mathematics, University of Maine, Orono, ME 04469, USA

2. 

Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

3. 

Department of Mathematics, Cornell University, Ithaca, NY 14850, USA

4. 

Department of Mathematics, University of North Georgia, Dahlonega, GA 30533, USA

Received  September 2018 Revised  December 2018 Published  December 2019 Early access  July 2019

The generalized Hunter-Saxton system comprises several well-kno-wn models from fluid dynamics and serves as a tool for the study of fluid convection and stretching in one-dimensional evolution equations. In this work, we examine the global regularity of periodic smooth solutions of this system in $ L^p $, $ p \in [1,\infty) $, spaces for nonzero real parameters $ (\lambda,\kappa) $. Our results significantly improve and extend those by Wunsch et al. [29,30,31] and Sarria [23]. Furthermore, we study the effects that different boundary conditions have on the global regularity of solutions by replacing periodicity with a homogeneous three-point boundary condition and establish finite-time blowup of a local-in-time solution of the resulting system for particular values of the parameters.

Citation: Jaeho Choi, Nitin Krishna, Nicole Magill, Alejandro Sarria. On the $ L^p $ regularity of solutions to the generalized Hunter-Saxton system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6349-6365. doi: 10.3934/dcdsb.2019142
References:
[1]

E. W. Barnes, A New Development of the Theory of Hypergeometric Functions, Proc. London Math. Soc., 6 (1908), 141-177.  doi: 10.1112/plms/s2-6.1.141.

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

S. ChildressG.R. IerleyE.A. Spiegel and W.R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech., 203 (1989), 1-22.  doi: 10.1017/S0022112089001357.

[4]

A. Constantin and R.I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050.

[5]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. An., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[6]

H.R. DullinG.A. Gottwald and D.D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dyn. Res., 33 (2003), 73-95.  doi: 10.1016/S0169-5983(03)00046-7.

[7]

A. ErdelyiW. MagnusF. Oberhettinger and F.G. Tricomi, Higher transcendental functions, Vol I, McGraw-Hill, 36 (1981), 56-119. 

[8]

J. EscherO. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Cont. Dyn. S., 19 (2007), 493-513.  doi: 10.3934/dcds.2007.19.493.

[9]

G. Gasper and M. Rahman, Basic Hypergeometric Series, Second edition. Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251.

[10]

A.V. Gurevich and K.P. Zybin, Nondissipative gravitational turbulence, Soviet Phys. JETP, 67 (1988), 1-12. 

[11]

A.V. Gurevich and K.P. Zybin, Large-scale structure of the Universe, Analytic theory, Soviet Phys. Usp., 38 (1995), 687-722. 

[12]

D.D. Holm and M.F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.

[13]

J.K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075.

[14]

R.S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.

[15]

J. Lenells and O. Lechtenfeld, On the $N = 2$ supersymmetric Camassa-Holm and Hunter-Saxton equations, J. Math. Phys., 50 (2009), 012704, 17 pp. doi: 10.1063/1.3060125.

[16]

J. Lenells and M. Wunsch, The Hunter-Saxton system and the geodesics on a pseudosphere, Commun. Part. Diff. Eq., 38 (2013), 860-881.  doi: 10.1080/03605302.2013.771660.

[17]

J. Liu and Z. Yin, Global weak solutions for a periodic two-component $\mu$-Hunter-Saxton system, Monatsh. Math., 168 (2012), 503-521.  doi: 10.1007/s00605-011-0346-9.

[18]

B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system, J. Differ. Equations, 253 (2012), 319-355.  doi: 10.1016/j.jde.2012.02.011.

[19]

B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system, Nonlinear Anal-Theor., 89 (2013), 242-249.  doi: 10.1016/j.na.2013.05.004.

[20]

H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Taiwan J. Math., 4 (2000), 65-103.  doi: 10.11650/twjm/1500407199.

[21]

M. V. Pavlov, The Gurevich-Zybin system, J. Phys. A-Math. Gen., 38 (2005), 3823-3840.  doi: 10.1088/0305-4470/38/17/008.

[22]

I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12 (1962), 161-168.  doi: 10.1017/S0022112062000130.

[23]

A. Sarria, Global estimates and blow-up criteria for the generalized Hunter-Saxton system, Discrete Cont. Dyn-B, 20 (2015), 641-673.  doi: 10.3934/dcdsb.2015.20.641.

[24]

A. Sarria, Regularity of stagnation-point form solutions of the two-dimensional Euler equations, Differential and Integral Equations, 28 (2015), 239-254. 

[25]

A. Sarria and R. Saxton, Blow-up of solutions to the generalized inviscid Proudman-Johnson equation, J. Math. Fluid Mech., 15 (2013), 493-523.  doi: 10.1007/s00021-012-0126-x.

[26]

A. Sarria and R. Saxton, The role of initial curvature in solutions to the generalized inviscid Proudman-Johnson equation, Quart. Appl. Math., 73 (2015), 55-91.  doi: 10.1090/S0033-569X-2015-01378-3.

[27]

R. Saxton and F. Tiglay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal., 40 (2008), 1499-1515.  doi: 10.1137/080713768.

[28]

M. Wunsch, The generalized Proudman-Johnson equation revisited, J. Math. Fluid Mech., 13 (2011), 147-154.  doi: 10.1007/s00021-009-0004-3.

[29]

M. Wunsch, On the Hunter-Saxton system, Discrete Cont. Dyn-B, 12 (2009), 647-656.  doi: 10.3934/dcdsb.2009.12.647.

[30]

M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304.  doi: 10.1137/090768576.

[31]

H. Wu and M. Wunsch, Global existence for the generalized two-component Hunter-Saxton system, J. Math. Fluid Mech., 14 (2012), 455-469.  doi: 10.1007/s00021-011-0075-9.

show all references

References:
[1]

E. W. Barnes, A New Development of the Theory of Hypergeometric Functions, Proc. London Math. Soc., 6 (1908), 141-177.  doi: 10.1112/plms/s2-6.1.141.

[2]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

S. ChildressG.R. IerleyE.A. Spiegel and W.R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech., 203 (1989), 1-22.  doi: 10.1017/S0022112089001357.

[4]

A. Constantin and R.I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050.

[5]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. An., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[6]

H.R. DullinG.A. Gottwald and D.D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dyn. Res., 33 (2003), 73-95.  doi: 10.1016/S0169-5983(03)00046-7.

[7]

A. ErdelyiW. MagnusF. Oberhettinger and F.G. Tricomi, Higher transcendental functions, Vol I, McGraw-Hill, 36 (1981), 56-119. 

[8]

J. EscherO. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Cont. Dyn. S., 19 (2007), 493-513.  doi: 10.3934/dcds.2007.19.493.

[9]

G. Gasper and M. Rahman, Basic Hypergeometric Series, Second edition. Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251.

[10]

A.V. Gurevich and K.P. Zybin, Nondissipative gravitational turbulence, Soviet Phys. JETP, 67 (1988), 1-12. 

[11]

A.V. Gurevich and K.P. Zybin, Large-scale structure of the Universe, Analytic theory, Soviet Phys. Usp., 38 (1995), 687-722. 

[12]

D.D. Holm and M.F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.  doi: 10.1137/S1111111102410943.

[13]

J.K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075.

[14]

R.S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.

[15]

J. Lenells and O. Lechtenfeld, On the $N = 2$ supersymmetric Camassa-Holm and Hunter-Saxton equations, J. Math. Phys., 50 (2009), 012704, 17 pp. doi: 10.1063/1.3060125.

[16]

J. Lenells and M. Wunsch, The Hunter-Saxton system and the geodesics on a pseudosphere, Commun. Part. Diff. Eq., 38 (2013), 860-881.  doi: 10.1080/03605302.2013.771660.

[17]

J. Liu and Z. Yin, Global weak solutions for a periodic two-component $\mu$-Hunter-Saxton system, Monatsh. Math., 168 (2012), 503-521.  doi: 10.1007/s00605-011-0346-9.

[18]

B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system, J. Differ. Equations, 253 (2012), 319-355.  doi: 10.1016/j.jde.2012.02.011.

[19]

B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system, Nonlinear Anal-Theor., 89 (2013), 242-249.  doi: 10.1016/j.na.2013.05.004.

[20]

H. Okamoto and J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Taiwan J. Math., 4 (2000), 65-103.  doi: 10.11650/twjm/1500407199.

[21]

M. V. Pavlov, The Gurevich-Zybin system, J. Phys. A-Math. Gen., 38 (2005), 3823-3840.  doi: 10.1088/0305-4470/38/17/008.

[22]

I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12 (1962), 161-168.  doi: 10.1017/S0022112062000130.

[23]

A. Sarria, Global estimates and blow-up criteria for the generalized Hunter-Saxton system, Discrete Cont. Dyn-B, 20 (2015), 641-673.  doi: 10.3934/dcdsb.2015.20.641.

[24]

A. Sarria, Regularity of stagnation-point form solutions of the two-dimensional Euler equations, Differential and Integral Equations, 28 (2015), 239-254. 

[25]

A. Sarria and R. Saxton, Blow-up of solutions to the generalized inviscid Proudman-Johnson equation, J. Math. Fluid Mech., 15 (2013), 493-523.  doi: 10.1007/s00021-012-0126-x.

[26]

A. Sarria and R. Saxton, The role of initial curvature in solutions to the generalized inviscid Proudman-Johnson equation, Quart. Appl. Math., 73 (2015), 55-91.  doi: 10.1090/S0033-569X-2015-01378-3.

[27]

R. Saxton and F. Tiglay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal., 40 (2008), 1499-1515.  doi: 10.1137/080713768.

[28]

M. Wunsch, The generalized Proudman-Johnson equation revisited, J. Math. Fluid Mech., 13 (2011), 147-154.  doi: 10.1007/s00021-009-0004-3.

[29]

M. Wunsch, On the Hunter-Saxton system, Discrete Cont. Dyn-B, 12 (2009), 647-656.  doi: 10.3934/dcdsb.2009.12.647.

[30]

M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304.  doi: 10.1137/090768576.

[31]

H. Wu and M. Wunsch, Global existence for the generalized two-component Hunter-Saxton system, J. Math. Fluid Mech., 14 (2012), 455-469.  doi: 10.1007/s00021-011-0075-9.

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