# American Institute of Mathematical Sciences

July  2011, 10(4): 1011-1036. doi: 10.3934/cpaa.2011.10.1011

## On the structure of solutions of nonlinear hyperbolic systems of conservation laws

 1 Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom 2 Department of Mathematics, Purdue University, 150 N. University Street 47907-2067, United States

Received  March 2010 Revised  December 2010 Published  April 2011

We are concerned with entropy solutions $u$ in $L^\infty$ of nonlinear hyperbolic systems of conservation laws. It is shown that, given any entropy function $\eta$ and any hyperplane $t=const.$, if $u$ satisfies a vanishing mean oscillation property on the half balls, then $\eta(u)$ has a trace $H^d$-almost everywhere on the hyperplane. For the general case, given any set $E$ of finite perimeter and its inner unit normal $\nu: \partial^*E \to S^d$ and assuming the vanishing mean oscillation property of $u$ on the half balls, we show that the weak trace of the vector field $(\eta(u), q(u))$, defined in Chen-Torres-Ziemer [9], satisfies a stronger property for any entropy pair $(\eta, q)$. We then introduce an approach to analyze the structure of bounded entropy solutions for the isentropic Euler equations.
Citation: Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011
##### References:
 [1] L. Ambrosio, G. Crippa and S. Maniglia, Traces and fine properties of a $BD$ class of vector fields and applications,, Ann. Fac. Sci. Toulouse Math., 14 (2005), 527. [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford University Press, (2000). [3] G. Anzellotti, Pairings between measures and functions and compensated compactness,, Ann. Mat. Pura Appl., 135 (1983), 293. doi: doi:10.1007/BF01781073. [4] G.-Q. Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III),, Acta Math. Sci., 6 (1986), 75. [5] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89. doi: doi:10.1007/s002050050146. [6] G.-Q. Chen and Ph. LeFloch, Compressible Euler equations with general pressure law,, Arch. Ration. Mech. Anal., 153 (2000), 221. doi: doi:10.1007/s00205-002-0229-2. [7] G.-Q. Chen and M. Rascle, Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 153 (2000), 205. doi: doi:10.1007/s002050000081. [8] G.-Q. Chen and M. Torres, Divergence-measure fields, sets of finite perimeter, and conservation laws,, Arch. Ration. Mech. Anal., 175 (2005), 245. doi: doi:10.1007/s00205-004-0346-1. [9] G.-Q. Chen, M. Torres and W. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws,, Comm. Pure Appl. Math., 62 (2009), 242. doi: doi:10.1002/cpa.20262. [10] G.-Q. Chen, M. Torres and W. P. Ziemer, Measure-theoretical analysis and nonlinear conservation laws,, Pure Appl. Math. Quarterly, 3 (2007), 841. [11] C. De Lellis, F. Otto and M. Westdickenberg, Structure of entropy solutions for multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 170 (2003), 137. doi: doi:10.1007/s00205-003-0270-9. [12] X. Ding, G.-Q. Chen and P. Luo, Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I)-(II),, Acta Math. Sci., 5 (1985), 483. [13] X. Ding, G.-Q. Chen and P. Luo, Convergence of the fractional step Lax-Friedrichs and Godunov scheme for the isentropic system of gas dynamics,, Commun. Math. Phys., 121 (1989), 63. doi: doi:10.1007/BF01218624. [14] R. J. DiPerna, Convergence of approximate solutions to conservation laws,, Arch. Ration. Mech. Anal., 82 (1983), 27. doi: doi:10.1007/BF00251724. [15] R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics,, Commun. Math. Phys., 91 (1983), 1. doi: doi:10.1007/BF01206047. [16] L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", CRC Press, (1992). [17] H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969). [18] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Birkh\, (1984). [19] P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation for the isentropic gas dynamics and p-system,, Commun. Math. Phys., 163 (1994), 415. doi: doi:10.1007/BF02102014. [20] P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates,, Comm. Pure Appl. Math., 49 (1996), 599. doi: doi:10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5. [21] E. Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws,, J. Hyper. Diff. Eqs., 2 (2005), 885. doi: doi:10.1142/S0219891605000658. [22] N. C. Phuc and M. Torres, Characterizations of the existence and removable singularities of divergence-measure vector fields,, Indiana Univ. Math. J., 57 (2008), 1573. doi: doi:10.1512/iumj.2008.57.3312. [23] W. Rudin, "Principi di Analisi Matematica,", McGraw-Hill, (1991). [24] M. Silhavy, Divergence measure fields and Cauchy's stress theorem,, Rend. Sem. Mat. Padova, 113 (2005), 15. [25] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181. doi: doi:10.1007/s002050100157. [26] A. Vasseur and Y. Kwon, Strong traces for solutions to scalar conservation laws with general flux,, Arch. Ration. Mech. Anal., 185 (2007), 495. doi: doi:10.1007/s00205-007-0055-7. [27] W. Ziemer, "Weakly Differentiable Functions,", Springer-Verlag, (1989).

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##### References:
 [1] L. Ambrosio, G. Crippa and S. Maniglia, Traces and fine properties of a $BD$ class of vector fields and applications,, Ann. Fac. Sci. Toulouse Math., 14 (2005), 527. [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford University Press, (2000). [3] G. Anzellotti, Pairings between measures and functions and compensated compactness,, Ann. Mat. Pura Appl., 135 (1983), 293. doi: doi:10.1007/BF01781073. [4] G.-Q. Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III),, Acta Math. Sci., 6 (1986), 75. [5] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89. doi: doi:10.1007/s002050050146. [6] G.-Q. Chen and Ph. LeFloch, Compressible Euler equations with general pressure law,, Arch. Ration. Mech. Anal., 153 (2000), 221. doi: doi:10.1007/s00205-002-0229-2. [7] G.-Q. Chen and M. Rascle, Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 153 (2000), 205. doi: doi:10.1007/s002050000081. [8] G.-Q. Chen and M. Torres, Divergence-measure fields, sets of finite perimeter, and conservation laws,, Arch. Ration. Mech. Anal., 175 (2005), 245. doi: doi:10.1007/s00205-004-0346-1. [9] G.-Q. Chen, M. Torres and W. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws,, Comm. Pure Appl. Math., 62 (2009), 242. doi: doi:10.1002/cpa.20262. [10] G.-Q. Chen, M. Torres and W. P. Ziemer, Measure-theoretical analysis and nonlinear conservation laws,, Pure Appl. Math. Quarterly, 3 (2007), 841. [11] C. De Lellis, F. Otto and M. Westdickenberg, Structure of entropy solutions for multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 170 (2003), 137. doi: doi:10.1007/s00205-003-0270-9. [12] X. Ding, G.-Q. Chen and P. Luo, Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I)-(II),, Acta Math. Sci., 5 (1985), 483. [13] X. Ding, G.-Q. Chen and P. Luo, Convergence of the fractional step Lax-Friedrichs and Godunov scheme for the isentropic system of gas dynamics,, Commun. Math. Phys., 121 (1989), 63. doi: doi:10.1007/BF01218624. [14] R. J. DiPerna, Convergence of approximate solutions to conservation laws,, Arch. Ration. Mech. Anal., 82 (1983), 27. doi: doi:10.1007/BF00251724. [15] R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics,, Commun. Math. Phys., 91 (1983), 1. doi: doi:10.1007/BF01206047. [16] L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", CRC Press, (1992). [17] H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969). [18] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Birkh\, (1984). [19] P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation for the isentropic gas dynamics and p-system,, Commun. Math. Phys., 163 (1994), 415. doi: doi:10.1007/BF02102014. [20] P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates,, Comm. Pure Appl. Math., 49 (1996), 599. doi: doi:10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5. [21] E. Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws,, J. Hyper. Diff. Eqs., 2 (2005), 885. doi: doi:10.1142/S0219891605000658. [22] N. C. Phuc and M. Torres, Characterizations of the existence and removable singularities of divergence-measure vector fields,, Indiana Univ. Math. J., 57 (2008), 1573. doi: doi:10.1512/iumj.2008.57.3312. [23] W. Rudin, "Principi di Analisi Matematica,", McGraw-Hill, (1991). [24] M. Silhavy, Divergence measure fields and Cauchy's stress theorem,, Rend. Sem. Mat. Padova, 113 (2005), 15. [25] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181. doi: doi:10.1007/s002050100157. [26] A. Vasseur and Y. Kwon, Strong traces for solutions to scalar conservation laws with general flux,, Arch. Ration. Mech. Anal., 185 (2007), 495. doi: doi:10.1007/s00205-007-0055-7. [27] W. Ziemer, "Weakly Differentiable Functions,", Springer-Verlag, (1989).
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