Examples and conjectures on the regularity of solutions to balance laws
Authors:
Fabio Ancona, Stefano Bianchini, Alberto Bressan, Rinaldo M. Colombo and Khai T. Nguyen
Journal:
Quart. Appl. Math. 81 (2023), 433-454
MSC (2020):
Primary 35L65, 35L67; Secondary 35L40
DOI:
https://doi.org/10.1090/qam/1647
Published electronically:
February 8, 2023
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Additional Information
Abstract: The paper discusses various regularity properties for solutions to a scalar, 1-dimensional conservation law with strictly convex flux and integrable source. In turn, these yield compactness estimates on the solution set. Similar properties are expected to hold for $2\times 2$ genuinely nonlinear systems.
References
- F. Ancona, O. Glass, and K. T. Nguyen, Lower compactness estimates for scalar balance laws, Comm. Pure Appl. Math. 65 (2012), 1303–1329.
- P. Baiti, A. Bressan, and H. K. Jenssen, BV instability of the Godunov scheme, Comm. Pure Appl. Math. 59 (2006), 1604–1638.
- P. Bénilan and M. Crandall, Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, Johns Hopkins Univ. Press, Baltimore, MD, 1981, pp. 23–39.
- S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Annals Math. 161 (2005), 223–342.
- S. Bianchini, R. M. Colombo, and F. Monti, $2\times 2$ systems of conservation laws with $L^\infty$ data, J. Differential Equations 249 (2010), 3466–3488.
- A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), 205–230.
- A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford University Press, Oxford, 2000.
- A. Bressan, Hyperbolic conservation laws: an illustrated tutorial, Modeling and Optimization of Flows on Networks, edited by L. Ambrosio, A. Bressan, D. Helbing, A. Klar, and E. Zuazua, Springer Lecture Notes in Mathematics 2062 (2012), pp. 157–245.
- A. Bressan, G. Chen, and Q. Zhang, On finite time BV blow-up for the $p$-system, Comm. Partial Diff. Equat. 43 (2018), 1242–1280.
- A. Bressan and R. M. Colombo, The semigroup generated by $2\times 2$ conservation laws, Arch. Rational Mech. Anal. 113 (1995), 1–75.
- A. Bressan and R. M. Colombo, Unique solutions of $2\times 2$ conservation laws with large data, Indiana Univ. Math. J. 44 (1995), 677–725.
- A. Bressan, G. Crasta, and B. Piccoli, Well posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Amer. Math. Soc. Memoir 694 (2000).
- A. Bressan and P. Goatin, Stability of $\mathbf {L}^\infty$ solutions of Temple class systems, Diff. Integ. Equat. 13 (2000), 1503–1528.
- A. Bressan, G. Guerra, and W. Shen, Vanishing viscosity solutions for conservation laws with regulated flux, J. Differential Equations 299 (2019), 312–351.
- A. Bressan, T. P. Liu, and T. Yang, ${\mathbf {L}}^1$ stability estimates for $n\times n$ conservation laws, Arch. Rational Mech. Anal. 149 (1999), 1–22.
- A. Bressan and K. Nguyen, Global existence of weak solutions for the Burgers-Hilbert equation, SIAM J. Math. Anal. 46 (2014), 2884–2904.
- A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B (7) 2 (1988), 641–656
- G. Q. Chen and M. Torres, On the structure of solutions of nonlinear hyperbolic systems of conservation laws, Comm. Pure Appl. Anal. 10 (2011), 1011–1036.
- M. G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132.
- M. G. Crandall, A generalized domain for semigroup generators, Proc. Amer. Math. Soc. 37 (1973), 434–440.
- M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298.
- C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33-41.
- C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer-Verlag, Berlin, 2016.
- G. Dal Maso, P. LeFloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995), 483–548.
- G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739–765.
- C. De Lellis and F. Golse, A quantitative compactness estimate for scalar conservation laws, Comm. Pure Appl. Math. 58 (2005), 989–998.
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27–70.
- J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715.
- J. Glimm and P. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society 101, Providence, RI, 1970.
- D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer, Berlin, 1981.
- H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, Berlin, 2002.
- D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc. 289 (1985), 591–610.
- H. K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal. 31 (2000), 894–908.
- S. Kruzhkov, First-order quasilinear equations with several space variables, Mat. Sb. 123 (1970), 228–255. English transl. Math. USSR Sb. 10 (1970), 217–273.
- M. Lewicka, Stability conditions for patterns of non-interacting large shock waves, SIAM J. Math. Anal. 32 (2001), 1094–1116.
- M. Lewicka, The well posedness for hyperbolic systems of conservation laws with large BV data, Arch. Rational Mech. Anal. 173 (2004), 415–445.
- A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser, Basel, 1995.
References
- F. Ancona, O. Glass, and K. T. Nguyen, Lower compactness estimates for scalar balance laws, Comm. Pure Appl. Math. 65 (2012), 1303–1329.
- P. Baiti, A. Bressan, and H. K. Jenssen, BV instability of the Godunov scheme, Comm. Pure Appl. Math. 59 (2006), 1604–1638.
- P. Bénilan and M. Crandall, Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, Johns Hopkins Univ. Press, Baltimore, MD, 1981, pp. 23–39.
- S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Annals Math. 161 (2005), 223–342.
- S. Bianchini, R. M. Colombo, and F. Monti, $2\times 2$ systems of conservation laws with $L^\infty$ data, J. Differential Equations 249 (2010), 3466–3488.
- A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), 205–230.
- A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford University Press, Oxford, 2000.
- A. Bressan, Hyperbolic conservation laws: an illustrated tutorial, Modeling and Optimization of Flows on Networks, edited by L. Ambrosio, A. Bressan, D. Helbing, A. Klar, and E. Zuazua, Springer Lecture Notes in Mathematics 2062 (2012), pp. 157–245.
- A. Bressan, G. Chen, and Q. Zhang, On finite time BV blow-up for the $p$-system, Comm. Partial Diff. Equat. 43 (2018), 1242–1280.
- A. Bressan and R. M. Colombo, The semigroup generated by $2\times 2$ conservation laws, Arch. Rational Mech. Anal. 113 (1995), 1–75.
- A. Bressan and R. M. Colombo, Unique solutions of $2\times 2$ conservation laws with large data, Indiana Univ. Math. J. 44 (1995), 677–725.
- A. Bressan, G. Crasta, and B. Piccoli, Well posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Amer. Math. Soc. Memoir 694 (2000).
- A. Bressan and P. Goatin, Stability of $\mathbf {L}^\infty$ solutions of Temple class systems, Diff. Integ. Equat. 13 (2000), 1503–1528.
- A. Bressan, G. Guerra, and W. Shen, Vanishing viscosity solutions for conservation laws with regulated flux, J. Differential Equations 299 (2019), 312–351.
- A. Bressan, T. P. Liu, and T. Yang, ${\mathbf {L}}^1$ stability estimates for $n\times n$ conservation laws, Arch. Rational Mech. Anal. 149 (1999), 1–22.
- A. Bressan and K. Nguyen, Global existence of weak solutions for the Burgers-Hilbert equation, SIAM J. Math. Anal. 46 (2014), 2884–2904.
- A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B (7) 2 (1988), 641–656
- G. Q. Chen and M. Torres, On the structure of solutions of nonlinear hyperbolic systems of conservation laws, Comm. Pure Appl. Anal. 10 (2011), 1011–1036.
- M. G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132.
- M. G. Crandall, A generalized domain for semigroup generators, Proc. Amer. Math. Soc. 37 (1973), 434–440.
- M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298.
- C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33-41.
- C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer-Verlag, Berlin, 2016.
- G. Dal Maso, P. LeFloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995), 483–548.
- G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739–765.
- C. De Lellis and F. Golse, A quantitative compactness estimate for scalar conservation laws, Comm. Pure Appl. Math. 58 (2005), 989–998.
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27–70.
- J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715.
- J. Glimm and P. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society 101, Providence, RI, 1970.
- D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer, Berlin, 1981.
- H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, Berlin, 2002.
- D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc. 289 (1985), 591–610.
- H. K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal. 31 (2000), 894–908.
- S. Kruzhkov, First-order quasilinear equations with several space variables, Mat. Sb. 123 (1970), 228–255. English transl. Math. USSR Sb. 10 (1970), 217–273.
- M. Lewicka, Stability conditions for patterns of non-interacting large shock waves, SIAM J. Math. Anal. 32 (2001), 1094–1116.
- M. Lewicka, The well posedness for hyperbolic systems of conservation laws with large BV data, Arch. Rational Mech. Anal. 173 (2004), 415–445.
- A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser, Basel, 1995.
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Additional Information
Fabio Ancona
Affiliation:
Dipartimento di Matematica “Tullio Levi-Civita” University of Padova, Via Trieste 63, 35121 Padova, Italy
MR Author ID:
292312
ORCID:
0000-0002-2686-5470
Email:
ancona@math.unipd.it
Stefano Bianchini
Affiliation:
SISSA, Via Bonomea 265, 34136 Trieste, Italy
MR Author ID:
621739
ORCID:
0000-0001-7886-3285
Email:
bianchin@sissa.it
Alberto Bressan
Affiliation:
Department of Mathematics, Penn State University, University Park, PA 16802, USA
MR Author ID:
191251
Email:
axb62@psu.edu
Rinaldo M. Colombo
Affiliation:
INdAM Unit & Department of Information Engineering, University of Brescia, Via Branze 38, 25123 Brescia, Italy
MR Author ID:
315089
ORCID:
0000-0003-0459-585X
Email:
rinaldo.colombo@unibs.it
Khai T. Nguyen
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
MR Author ID:
887358
ORCID:
0000-0001-7821-3370
Email:
khai@math.ncsu.edu
Received by editor(s):
November 13, 2022
Published electronically:
February 8, 2023
Additional Notes:
The research of the first author was partially supported by the Istituto Nazionale di Alta Matematica ’F. Severi’, through GNAMPA. The research of the third author was partially supported by NSF with grant DMS-2006884 “Singularities and error bounds for hyperbolic equations”. The research of the fifth author was partially supported by NSF with grant DMS-2154201 “Generic Singularities and Fine Regularity Structure for Nonlinear Partial Differential Equations”. F. Ancona, S. Bianchini, R. M. Colombo and K. T. Nguyen acknowledge the hospitality of Penn State University, where the collaboration leading to this work took place.
Dedicated:
Dedicated to Constantine Dafermos, mentor and friend
Article copyright:
© Copyright 2023
Brown University
