Figure 1.
Proof of a rough Hölder continuity estimate of u
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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 |
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 ].
Keywords:
Continuous solutions,
entropy solutions,
balance laws,
Lagrangian formulation,
Hölder continuity.
Mathematics Subject Classification:
Primary: 35L60, 37C10, 58J45.
Citation:
Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure and 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. |
| [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. |
| [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. |
| [4] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Clarendon Press, 2000. |
| [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. |
| [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. |
| [7] |
S. Bianchini and E. Marconi, On the structure of L1-entropy solutions to scalar conservation laws in one space dimension, preprint. |
| [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. |
| [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. |
| [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. |
| [11] |
C. M. Dafermos, Continuous solutions for balance laws, Ric. Mat., 55 (2006), 79–91.
doi: 10.1007/s11587-006-0006-x. |
| [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. |
| [13] |
H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. |
| [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. |
| [15] |
J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag New York, 2001.
doi: 10.1007/978-1-4613-0131-8. |
| [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. |
| [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. |
| [18] |
S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb., 81 (1970), 228–255.
doi: 10.1070/SM1970v010n02ABEH002156. |
| [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. |
| [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. |
| [21] |
J. Von Neumann, On rings of operators: Reduction Theory, Ann. of Math. (2), 50 (1949), 401–485.
doi: 10.2307/1969463. |
| [22] |
S. M. Srivastava, A Course on Borel Sets, Grad. Texts Math. , vol. 180, Springer, 1998.
doi: 10.1007/978-3-642-85473-6. |
| [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. |
| [24] |
D. Vittone, Submanifolds in Carnot Groups, Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, Birkhaüser, 2008. |
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. |
| [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. |
| [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. |
| [4] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Clarendon Press, 2000. |
| [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. |
| [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. |
| [7] |
S. Bianchini and E. Marconi, On the structure of L1-entropy solutions to scalar conservation laws in one space dimension, preprint. |
| [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. |
| [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. |
| [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. |
| [11] |
C. M. Dafermos, Continuous solutions for balance laws, Ric. Mat., 55 (2006), 79–91.
doi: 10.1007/s11587-006-0006-x. |
| [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. |
| [13] |
H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. |
| [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. |
| [15] |
J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag New York, 2001.
doi: 10.1007/978-1-4613-0131-8. |
| [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. |
| [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. |
| [18] |
S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb., 81 (1970), 228–255.
doi: 10.1070/SM1970v010n02ABEH002156. |
| [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. |
| [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. |
| [21] |
J. Von Neumann, On rings of operators: Reduction Theory, Ann. of Math. (2), 50 (1949), 401–485.
doi: 10.2307/1969463. |
| [22] |
S. M. Srivastava, A Course on Borel Sets, Grad. Texts Math. , vol. 180, Springer, 1998.
doi: 10.1007/978-3-642-85473-6. |
| [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. |
| [24] |
D. Vittone, Submanifolds in Carnot Groups, Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, Birkhaüser, 2008. |
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