# American Institute of Mathematical Sciences

June  2016, 11(2): 349-367. doi: 10.3934/nhm.2016.11.349

## On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws

 1 Novgorod State University, 41 B. Sankt-Peterburgskaya, 173003 Veliky Novgorod, Russian Federation

Received  April 2015 Revised  June 2015 Published  March 2016

We propose a new sufficient non-degeneracy condition for the strong precompactness of bounded sequences satisfying the nonlinear first-order differential constraints. This result is applied to establish the decay property for periodic entropy solutions to multidimensional scalar conservation laws.
Citation: Evgeny Yu. Panov. On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws. Networks and Heterogeneous Media, 2016, 11 (2) : 349-367. doi: 10.3934/nhm.2016.11.349
##### References:
 [1] A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, 1932. [2] G.-Q. Chen and H. Frid, Decay of entropy solutions of nonlinear conservation laws, Arch. Rational Mech. Anal., 146 (1999), 95-127. doi: 10.1007/s002050050138. [3] G.-Q. Chen and Y.-G. Lu, The study on application way of the compensated compactness theory, Chinese Sci. Bull., 34 (1989), 15-19. [4] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1. [5] C. M. Dafermos, Long time behavior of periodic solutions to scalar conservation laws in several space dimensions, SIAM J. Math. Anal., 45 (2013), 2064-2070. doi: 10.1137/130909688. [6] R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112. [7] P. Gerárd, Microlocal defect measures, Comm. Partial Diff. Equat., 16 (1991), 1761-1794. doi: 10.1080/03605309108820822. [8] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Providence, 1957. [9] S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), 228-255, English transl. in Math. USSR Sb., 10 (1970), 217-243. [10] S. N. Kruzhkov and E. Yu. Panov, First-order conservative quasilinear laws with an infinite domain of dependence on the initial data, Dokl. Akad. Nauk SSSR, 314 (1990), 79-84, English transl. in Soviet Math. Dokl., 42 (1991), 316-321. [11] S. N. Kruzhkov and E. Yu. Panov, Osgood's type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order, Ann. Univ. Ferrara Sez. VII (N.S.), 40 (1994), 31-54. [12] S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media, Comp. GeoSci., 14 (2010), 105-124. doi: 10.1007/s10596-009-9135-0. [13] E. Yu. Panov, On sequences of measure-valued solutions of first-order quasilinear equations, Mat. Sb., 185 (1994), 87-106, English transl. in Russian Acad. Sci. Sb. Math., 81 (1995), 211-227. doi: 10.1070/SM1995v081n01ABEH003621. [14] E. Yu. Panov, Property of strong precompactness for bounded sets of measure valued solutions of a first-order quasilinear equation, Mat. Sb., 190 (1999), 109-128, English transl. in Russian Acad. Sci. Sb. Math., 190 (1999), 427-446. doi: 10.1070/SM1999v190n03ABEH000395. [15] E. Yu. Panov, A remark on the theory of generalized entropy sub- and supersolutions of the Cauchy problem for a first-order quasilinear equation, Differ. Uravn., 37 (2001), 252-259, English transl. in Differ. Equ., 37 (2001), 272-280. doi: 10.1023/A:1019273927768. [16] E. Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 885-908. doi: 10.1142/S0219891605000658. [17] E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343. [18] E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Rational Mech. Anal., 195 (2010), 643-673. doi: 10.1007/s00205-009-0217-x. [19] E. Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property, J. Math. Sci., 159 (2009), 180-228. doi: 10.1007/s10958-009-9434-y. [20] E. Yu. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law, SIAM J. Math. Anal., 41 (2009), 26-36. doi: 10.1137/080724587. [21] E. Yu. Panov, On decay of periodic entropy solutions to a scalar conservation law, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 997-1007. doi: 10.1016/j.anihpc.2012.12.009. [22] E. Yu. Panov, On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property, preprint,, , (). [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. [24] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics, Heriot. Watt Symposium, vol. 4 (Edinburgh 1979), Res. Notes Math., 39 (1979), 136-212. [25] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh. Sect. A., 115 (1990), 193-230. doi: 10.1017/S0308210500020606.

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##### References:
 [1] A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, 1932. [2] G.-Q. Chen and H. Frid, Decay of entropy solutions of nonlinear conservation laws, Arch. Rational Mech. Anal., 146 (1999), 95-127. doi: 10.1007/s002050050138. [3] G.-Q. Chen and Y.-G. Lu, The study on application way of the compensated compactness theory, Chinese Sci. Bull., 34 (1989), 15-19. [4] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1. [5] C. M. Dafermos, Long time behavior of periodic solutions to scalar conservation laws in several space dimensions, SIAM J. Math. Anal., 45 (2013), 2064-2070. doi: 10.1137/130909688. [6] R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112. [7] P. Gerárd, Microlocal defect measures, Comm. Partial Diff. Equat., 16 (1991), 1761-1794. doi: 10.1080/03605309108820822. [8] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Providence, 1957. [9] S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), 228-255, English transl. in Math. USSR Sb., 10 (1970), 217-243. [10] S. N. Kruzhkov and E. Yu. Panov, First-order conservative quasilinear laws with an infinite domain of dependence on the initial data, Dokl. Akad. Nauk SSSR, 314 (1990), 79-84, English transl. in Soviet Math. Dokl., 42 (1991), 316-321. [11] S. N. Kruzhkov and E. Yu. Panov, Osgood's type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order, Ann. Univ. Ferrara Sez. VII (N.S.), 40 (1994), 31-54. [12] S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media, Comp. GeoSci., 14 (2010), 105-124. doi: 10.1007/s10596-009-9135-0. [13] E. Yu. Panov, On sequences of measure-valued solutions of first-order quasilinear equations, Mat. Sb., 185 (1994), 87-106, English transl. in Russian Acad. Sci. Sb. Math., 81 (1995), 211-227. doi: 10.1070/SM1995v081n01ABEH003621. [14] E. Yu. Panov, Property of strong precompactness for bounded sets of measure valued solutions of a first-order quasilinear equation, Mat. Sb., 190 (1999), 109-128, English transl. in Russian Acad. Sci. Sb. Math., 190 (1999), 427-446. doi: 10.1070/SM1999v190n03ABEH000395. [15] E. Yu. Panov, A remark on the theory of generalized entropy sub- and supersolutions of the Cauchy problem for a first-order quasilinear equation, Differ. Uravn., 37 (2001), 252-259, English transl. in Differ. Equ., 37 (2001), 272-280. doi: 10.1023/A:1019273927768. [16] E. Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 885-908. doi: 10.1142/S0219891605000658. [17] E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343. [18] E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Rational Mech. Anal., 195 (2010), 643-673. doi: 10.1007/s00205-009-0217-x. [19] E. Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property, J. Math. Sci., 159 (2009), 180-228. doi: 10.1007/s10958-009-9434-y. [20] E. Yu. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law, SIAM J. Math. Anal., 41 (2009), 26-36. doi: 10.1137/080724587. [21] E. Yu. Panov, On decay of periodic entropy solutions to a scalar conservation law, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 997-1007. doi: 10.1016/j.anihpc.2012.12.009. [22] E. Yu. Panov, On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property, preprint,, , (). [23] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. [24] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics, Heriot. Watt Symposium, vol. 4 (Edinburgh 1979), Res. Notes Math., 39 (1979), 136-212. [25] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh. Sect. A., 115 (1990), 193-230. doi: 10.1017/S0308210500020606.
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