# 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.   Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford University Press, (2000).   Google Scholar [3] G. Anzellotti, Pairings between measures and functions and compensated compactness,, Ann. Mat. Pura Appl., 135 (1983), 293.  doi: doi:10.1007/BF01781073.  Google Scholar [4] G.-Q. Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III),, Acta Math. Sci., 6 (1986), 75.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] G.-Q. Chen, M. Torres and W. P. Ziemer, Measure-theoretical analysis and nonlinear conservation laws,, Pure Appl. Math. Quarterly, 3 (2007), 841.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [14] R. J. DiPerna, Convergence of approximate solutions to conservation laws,, Arch. Ration. Mech. Anal., 82 (1983), 27.  doi: doi:10.1007/BF00251724.  Google Scholar [15] R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics,, Commun. Math. Phys., 91 (1983), 1.  doi: doi:10.1007/BF01206047.  Google Scholar [16] L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", CRC Press, (1992).   Google Scholar [17] H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969).   Google Scholar [18] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Birkh\, (1984).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] W. Rudin, "Principi di Analisi Matematica,", McGraw-Hill, (1991).   Google Scholar [24] M. Silhavy, Divergence measure fields and Cauchy's stress theorem,, Rend. Sem. Mat. Padova, 113 (2005), 15.   Google Scholar [25] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181.  doi: doi:10.1007/s002050100157.  Google Scholar [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.  Google Scholar [27] W. Ziemer, "Weakly Differentiable Functions,", Springer-Verlag, (1989).   Google Scholar

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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.   Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford University Press, (2000).   Google Scholar [3] G. Anzellotti, Pairings between measures and functions and compensated compactness,, Ann. Mat. Pura Appl., 135 (1983), 293.  doi: doi:10.1007/BF01781073.  Google Scholar [4] G.-Q. Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III),, Acta Math. Sci., 6 (1986), 75.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] G.-Q. Chen, M. Torres and W. P. Ziemer, Measure-theoretical analysis and nonlinear conservation laws,, Pure Appl. Math. Quarterly, 3 (2007), 841.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [14] R. J. DiPerna, Convergence of approximate solutions to conservation laws,, Arch. Ration. Mech. Anal., 82 (1983), 27.  doi: doi:10.1007/BF00251724.  Google Scholar [15] R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics,, Commun. Math. Phys., 91 (1983), 1.  doi: doi:10.1007/BF01206047.  Google Scholar [16] L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions,", CRC Press, (1992).   Google Scholar [17] H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969).   Google Scholar [18] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Birkh\, (1984).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] W. Rudin, "Principi di Analisi Matematica,", McGraw-Hill, (1991).   Google Scholar [24] M. Silhavy, Divergence measure fields and Cauchy's stress theorem,, Rend. Sem. Mat. Padova, 113 (2005), 15.   Google Scholar [25] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181.  doi: doi:10.1007/s002050100157.  Google Scholar [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.  Google Scholar [27] W. Ziemer, "Weakly Differentiable Functions,", Springer-Verlag, (1989).   Google Scholar
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