June  2016, 11(2): 263-280. doi: 10.3934/nhm.2016.11.263

New interaction estimates for the Baiti-Jenssen system

1. 

Dipartimento di Matematica, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy

2. 

IMATI-CNR, via Ferrata 1, 27100 Pavia, Italy

Received  April 2015 Revised  October 2015 Published  March 2016

We establish new interaction estimates for a system introduced by Baiti and Jenssen. These estimates are pivotal to the analysis of the wave front-tracking approximation. In a companion paper we use them to construct a counter-example which shows that Schaeffer's Regularity Theorem for scalar conservation laws does not extend to systems. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. The proof of the interaction estimates is based on the explicit computation of the wave fan curves and on a perturbation argument.
Citation: Laura Caravenna, Laura V. Spinolo. New interaction estimates for the Baiti-Jenssen system. Networks & Heterogeneous Media, 2016, 11 (2) : 263-280. doi: 10.3934/nhm.2016.11.263
References:
[1]

L. Ambrosio and C. De Lellis, A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations, J. Hyperbolic Differ. Equ., 1 (2004), 813-826. doi: 10.1142/S0219891604000263.  Google Scholar

[2]

F. Ancona and K. T. Nguyen, in, preparation., ().   Google Scholar

[3]

P. Baiti and H. K. Jenssen, Blowup in $L^\infty$ for a class of genuinely nonlinear hyperbolic systems of conservation laws, Discrete Contin. Dynam. Systems, 7 (2001), 837-853. doi: 10.3934/dcds.2001.7.837.  Google Scholar

[4]

S. Bianchini and L. Caravenna, SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension, Comm. Math. Phys., 313 (2012), 1-33. doi: 10.1007/s00220-012-1480-5.  Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000.  Google Scholar

[6]

A. Bressan and G. M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim., 41 (2002), 607-622. doi: 10.1137/S0363012901392529.  Google Scholar

[7]

A. Bressan and R. M. Colombo, Decay of positive waves in nonlinear systems of conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 133-160, URL http://www.numdam.org/item?id=ASNSP_1998_4_26_1_133_0.  Google Scholar

[8]

A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for $n\times n$ conservation laws, J. Differential Equations, 156 (1999), 26-49. doi: 10.1006/jdeq.1998.3606.  Google Scholar

[9]

A. Bressan and T. Yang, A sharp decay estimate for positive nonlinear waves, SIAM J. Math. Anal., 36 (2004), 659-677. doi: 10.1137/S0036141003427774.  Google Scholar

[10]

L. Caravenna, A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation, Bull. Braz. Math. Soc. (N.S.), 47 (1) (2016), Also arXiv:1505.00531. Google Scholar

[11]

L. Caravenna and L. V. Spinolo, Schaeffer's Regularity Theorem for scalar conservation laws does not extend to systems,, Indiana Univ. Math. J., ().   Google Scholar

[12]

C. Christoforou and K. Trivisa, Sharp decay estimates for hyperbolic balance laws, J. Differential Equations, 247 (2009), 401-423. doi: 10.1016/j.jde.2009.03.013.  Google Scholar

[13]

C. M. Dafermos, Wave fans are special, Acta Math. Appl. Sin. Engl. Ser., 24 (2008), 369-374. doi: 10.1007/s10255-008-8010-4.  Google Scholar

[14]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 2010, doi: 10.1007/978-3-642-04048-1.  Google Scholar

[15]

R. J. DiPerna, Global solutions to a class of nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 26 (1973), 1-28. doi: 10.1002/cpa.3160260102.  Google Scholar

[16]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715. doi: 10.1002/cpa.3160180408.  Google Scholar

[17]

J. Glimm and P. D. Lax, Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970.  Google Scholar

[18]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002, doi: 10.1007/978-3-642-56139-9.  Google Scholar

[19]

H. K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal., 31 (2000), 894-908. doi: 10.1137/S0036141099352339.  Google Scholar

[20]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (123) (1970), 228-255.  Google Scholar

[21]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[22]

T. P. Liu, Decay to $N$-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 586-611.  Google Scholar

[23]

O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Uspehi Mat. Nauk (N.S.), 12 (1957), 3-73.  Google Scholar

[24]

D. G. Schaeffer, A regularity theorem for conservation laws, Advances in Math., 11 (1973), 368-386. doi: 10.1016/0001-8708(73)90018-2.  Google Scholar

show all references

References:
[1]

L. Ambrosio and C. De Lellis, A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations, J. Hyperbolic Differ. Equ., 1 (2004), 813-826. doi: 10.1142/S0219891604000263.  Google Scholar

[2]

F. Ancona and K. T. Nguyen, in, preparation., ().   Google Scholar

[3]

P. Baiti and H. K. Jenssen, Blowup in $L^\infty$ for a class of genuinely nonlinear hyperbolic systems of conservation laws, Discrete Contin. Dynam. Systems, 7 (2001), 837-853. doi: 10.3934/dcds.2001.7.837.  Google Scholar

[4]

S. Bianchini and L. Caravenna, SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension, Comm. Math. Phys., 313 (2012), 1-33. doi: 10.1007/s00220-012-1480-5.  Google Scholar

[5]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000.  Google Scholar

[6]

A. Bressan and G. M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim., 41 (2002), 607-622. doi: 10.1137/S0363012901392529.  Google Scholar

[7]

A. Bressan and R. M. Colombo, Decay of positive waves in nonlinear systems of conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 133-160, URL http://www.numdam.org/item?id=ASNSP_1998_4_26_1_133_0.  Google Scholar

[8]

A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for $n\times n$ conservation laws, J. Differential Equations, 156 (1999), 26-49. doi: 10.1006/jdeq.1998.3606.  Google Scholar

[9]

A. Bressan and T. Yang, A sharp decay estimate for positive nonlinear waves, SIAM J. Math. Anal., 36 (2004), 659-677. doi: 10.1137/S0036141003427774.  Google Scholar

[10]

L. Caravenna, A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation, Bull. Braz. Math. Soc. (N.S.), 47 (1) (2016), Also arXiv:1505.00531. Google Scholar

[11]

L. Caravenna and L. V. Spinolo, Schaeffer's Regularity Theorem for scalar conservation laws does not extend to systems,, Indiana Univ. Math. J., ().   Google Scholar

[12]

C. Christoforou and K. Trivisa, Sharp decay estimates for hyperbolic balance laws, J. Differential Equations, 247 (2009), 401-423. doi: 10.1016/j.jde.2009.03.013.  Google Scholar

[13]

C. M. Dafermos, Wave fans are special, Acta Math. Appl. Sin. Engl. Ser., 24 (2008), 369-374. doi: 10.1007/s10255-008-8010-4.  Google Scholar

[14]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 2010, doi: 10.1007/978-3-642-04048-1.  Google Scholar

[15]

R. J. DiPerna, Global solutions to a class of nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 26 (1973), 1-28. doi: 10.1002/cpa.3160260102.  Google Scholar

[16]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715. doi: 10.1002/cpa.3160180408.  Google Scholar

[17]

J. Glimm and P. D. Lax, Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970.  Google Scholar

[18]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002, doi: 10.1007/978-3-642-56139-9.  Google Scholar

[19]

H. K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal., 31 (2000), 894-908. doi: 10.1137/S0036141099352339.  Google Scholar

[20]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (123) (1970), 228-255.  Google Scholar

[21]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[22]

T. P. Liu, Decay to $N$-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 586-611.  Google Scholar

[23]

O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Uspehi Mat. Nauk (N.S.), 12 (1957), 3-73.  Google Scholar

[24]

D. G. Schaeffer, A regularity theorem for conservation laws, Advances in Math., 11 (1973), 368-386. doi: 10.1016/0001-8708(73)90018-2.  Google Scholar

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