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March  2017, 16(2): 629-644. doi: 10.3934/cpaa.2017031

Regularity estimates for continuous solutions of α-convex balance laws

Dipartimento di Matematica 'Tullio Levi-Civita', Università degli Studi di Padova, via Trieste 63,35121 Padova, Italy

Received  August 2016 Revised  October 2016 Published  January 2017

This paper proves new regularity estimates for continuous solutions to the balance equation
${{\partial }_{t}}u+{{\partial }_{x}}f(u)=g\qquad g\ \text{bounded}, f\in {{C}^{\text{2}n}}(\mathbb{R})$
when the flux $f$ satisfies a convexity assumption that we denote as 2n-convexity. The results are known in the case of the quadratic flux by very different arguments in [14,10,8]. We prove that the continuity of $u$ must be in fact $1/2n$-Hölder continuity and that the distributional source term $g$ is determined by the classical derivative of $u$ along any characteristics; part of the proof consists in showing that this classical derivative is well defined at any `Lebesgue point' of $g$ for suitable coverings. These two regularity statements fail in general for $C^{\infty}(\mathbb{R})$, strictly convex fluxes, see [3].
Citation: Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure & Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031
References:
[1]

G. Alberti, S. Bianchini and L. Caravenna, Reduction on characteristics for continuous solution of a scalar balance law, in Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series on Applied Mathematics, 8 (2014), 399–406. Google Scholar

[2]

G. Alberti, S. Bianchini and L. Caravenna, Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux Ⅰ, J. Differential Equations, 261 (2016), 4298–4337. doi: 10.1016/j.jde.2016.06.026.  Google Scholar

[3]

G. Alberti, S. Bianchini and L. Caravenna, Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux Ⅱ, Preprint SISSA 32/2016/MATE. doi: 10.1016/j.jde.2016.06.026.  Google Scholar

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Clarendon Press, 2000.  Google Scholar

[5]

L. Ambrosio, F. Serra Cassano and D. Vittone, Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal., 16 (2006), 187–232. doi: 10.1007/BF02922114.  Google Scholar

[6]

S. Bianchini and L. Caravenna, On optimality of c-cyclically monotone transference plans, C. R. Math. Acad. Sci. Paris, Ser. Ⅰ, 348 (2010), 613–618. doi: 10.1016/j.crma.2010.03.022.  Google Scholar

[7]

S. Bianchini and E. Marconi, On the structure of L1-entropy solutions to scalar conservation laws in one space dimension, preprint. Google Scholar

[8]

F. Bigolin, L. Caravenna and F. Serra Cassano, Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation, Ann. Inst. H. Poincar´e Analyse Non Lin´eaire., 32 (2015), 925–963. doi: 10.1016/j.anihpc.2014.05.001.  Google Scholar

[9]

F. Bigolin and F. Serra Cassano, Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs, Adv. Calc. Var., 3 (2010), 69–97. doi: 10.1515/ACV.2010.004.  Google Scholar

[10]

G. Citti, M. Manfredini, A. Pinamonti and F. Serra Cassano, Smooth approximation for the intrinsic Lipschitz functions in the Heisenberg group, Calc. Var. Partial Differ. Equ., 49 (2014), 1279–1308. doi: 10.1007/s00526-013-0622-8.  Google Scholar

[11]

C. M. Dafermos, Continuous solutions for balance laws, Ric. Mat., 55 (2006), 79–91. doi: 10.1007/s11587-006-0006-x.  Google Scholar

[12]

E. De Giorgi, F. Colombini and L. C. Piccinini, Frontiere orientate di misura minima e questione collegate, Classe di Scienze, Scuola Normale Superiore, Pisa, 1972.  Google Scholar

[13]

H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.  Google Scholar

[14]

B. Franchi, R. Serapioni and F. Serra Cassano, Differentiability of intrinsic Lipschitz functions within Heisenberg groups, Springer-Verlag New York., 21 (2011), 1044–1084. doi: 10.1007/s12220-010-9178-4.  Google Scholar

[15]

J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4613-0131-8.  Google Scholar

[16]

H. Holden and R. Xavier, Global semigroup of conservative solutions of the nonlinear variational wave equation, Arch. Ration. Mech. Anal., 201 (2011), 871–964. doi: 10.1007/s00205-011-0403-5.  Google Scholar

[17]

B. Kirchheim and F. Serra Cassano, Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (2004), 871–896. doi: 10.2422/2036-2145.2004.4.07.  Google Scholar

[18]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb., 81 (1970), 228–255. doi: 10.1070/SM1970v010n02ABEH002156.  Google Scholar

[19]

K. Kunugui, Contributions à la théorie des ensembles boreliens et analytiques Ⅱ and Ⅲ, J. Fac. Sci. Hokkaido Imp. Univ. Ser. Ⅰ, 8 (1939), 79–108 and 8 (1940), 1–24. Google Scholar

[20]

R. Monti and D. Vittone, Sets with finite Hn-perimeter and controlled normal, Math. Z., 270 (2012), 351–367. doi: 10.1007/s00209-010-0801-7.  Google Scholar

[21]

J. Von Neumann, On rings of operators: Reduction Theory, Ann. of Math. (2), 50 (1949), 401–485. doi: 10.2307/1969463.  Google Scholar

[22]

S. M. Srivastava, A Course on Borel Sets, Grad. Texts Math. , vol. 180, Springer, 1998. doi: 10.1007/978-3-642-85473-6.  Google Scholar

[23]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press, 1993.  Google Scholar

[24]

D. Vittone, Submanifolds in Carnot Groups, Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, Birkhaüser, 2008.  Google Scholar

show all references

References:
[1]

G. Alberti, S. Bianchini and L. Caravenna, Reduction on characteristics for continuous solution of a scalar balance law, in Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series on Applied Mathematics, 8 (2014), 399–406. Google Scholar

[2]

G. Alberti, S. Bianchini and L. Caravenna, Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux Ⅰ, J. Differential Equations, 261 (2016), 4298–4337. doi: 10.1016/j.jde.2016.06.026.  Google Scholar

[3]

G. Alberti, S. Bianchini and L. Caravenna, Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux Ⅱ, Preprint SISSA 32/2016/MATE. doi: 10.1016/j.jde.2016.06.026.  Google Scholar

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Clarendon Press, 2000.  Google Scholar

[5]

L. Ambrosio, F. Serra Cassano and D. Vittone, Intrinsic regular hypersurfaces in Heisenberg groups, J. Geom. Anal., 16 (2006), 187–232. doi: 10.1007/BF02922114.  Google Scholar

[6]

S. Bianchini and L. Caravenna, On optimality of c-cyclically monotone transference plans, C. R. Math. Acad. Sci. Paris, Ser. Ⅰ, 348 (2010), 613–618. doi: 10.1016/j.crma.2010.03.022.  Google Scholar

[7]

S. Bianchini and E. Marconi, On the structure of L1-entropy solutions to scalar conservation laws in one space dimension, preprint. Google Scholar

[8]

F. Bigolin, L. Caravenna and F. Serra Cassano, Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation, Ann. Inst. H. Poincar´e Analyse Non Lin´eaire., 32 (2015), 925–963. doi: 10.1016/j.anihpc.2014.05.001.  Google Scholar

[9]

F. Bigolin and F. Serra Cassano, Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs, Adv. Calc. Var., 3 (2010), 69–97. doi: 10.1515/ACV.2010.004.  Google Scholar

[10]

G. Citti, M. Manfredini, A. Pinamonti and F. Serra Cassano, Smooth approximation for the intrinsic Lipschitz functions in the Heisenberg group, Calc. Var. Partial Differ. Equ., 49 (2014), 1279–1308. doi: 10.1007/s00526-013-0622-8.  Google Scholar

[11]

C. M. Dafermos, Continuous solutions for balance laws, Ric. Mat., 55 (2006), 79–91. doi: 10.1007/s11587-006-0006-x.  Google Scholar

[12]

E. De Giorgi, F. Colombini and L. C. Piccinini, Frontiere orientate di misura minima e questione collegate, Classe di Scienze, Scuola Normale Superiore, Pisa, 1972.  Google Scholar

[13]

H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.  Google Scholar

[14]

B. Franchi, R. Serapioni and F. Serra Cassano, Differentiability of intrinsic Lipschitz functions within Heisenberg groups, Springer-Verlag New York., 21 (2011), 1044–1084. doi: 10.1007/s12220-010-9178-4.  Google Scholar

[15]

J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4613-0131-8.  Google Scholar

[16]

H. Holden and R. Xavier, Global semigroup of conservative solutions of the nonlinear variational wave equation, Arch. Ration. Mech. Anal., 201 (2011), 871–964. doi: 10.1007/s00205-011-0403-5.  Google Scholar

[17]

B. Kirchheim and F. Serra Cassano, Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (2004), 871–896. doi: 10.2422/2036-2145.2004.4.07.  Google Scholar

[18]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb., 81 (1970), 228–255. doi: 10.1070/SM1970v010n02ABEH002156.  Google Scholar

[19]

K. Kunugui, Contributions à la théorie des ensembles boreliens et analytiques Ⅱ and Ⅲ, J. Fac. Sci. Hokkaido Imp. Univ. Ser. Ⅰ, 8 (1939), 79–108 and 8 (1940), 1–24. Google Scholar

[20]

R. Monti and D. Vittone, Sets with finite Hn-perimeter and controlled normal, Math. Z., 270 (2012), 351–367. doi: 10.1007/s00209-010-0801-7.  Google Scholar

[21]

J. Von Neumann, On rings of operators: Reduction Theory, Ann. of Math. (2), 50 (1949), 401–485. doi: 10.2307/1969463.  Google Scholar

[22]

S. M. Srivastava, A Course on Borel Sets, Grad. Texts Math. , vol. 180, Springer, 1998. doi: 10.1007/978-3-642-85473-6.  Google Scholar

[23]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press, 1993.  Google Scholar

[24]

D. Vittone, Submanifolds in Carnot Groups, Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, Birkhaüser, 2008.  Google Scholar

Figure 1.  Proof of a rough Hölder continuity estimate of u
Figure 2.  Balances on characteristic regions
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