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 & 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). Google Scholar

[2]

G.-Q. Chen and H. Frid, Decay of entropy solutions of nonlinear conservation laws,, Arch. Rational Mech. Anal., 146 (1999), 95. doi: 10.1007/s002050050138. Google Scholar

[3]

G.-Q. Chen and Y.-G. Lu, The study on application way of the compensated compactness theory,, Chinese Sci. Bull., 34 (1989), 15. Google Scholar

[4]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, 3rd edition, (2010). doi: 10.1007/978-3-642-04048-1. Google Scholar

[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. doi: 10.1137/130909688. Google Scholar

[6]

R. J. DiPerna, Measure-valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. doi: 10.1007/BF00752112. Google Scholar

[7]

P. Gerárd, Microlocal defect measures,, Comm. Partial Diff. Equat., 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar

[8]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, Providence, (1957). Google Scholar

[9]

S. N. Kruzhkov, First order quasilinear equations in several independent variables,, Mat. Sb., 81 (1970), 228. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media,, Comp. GeoSci., 14 (2010), 105. doi: 10.1007/s10596-009-9135-0. Google Scholar

[13]

E. Yu. Panov, On sequences of measure-valued solutions of first-order quasilinear equations,, Mat. Sb., 185 (1994), 87. doi: 10.1070/SM1995v081n01ABEH003621. Google Scholar

[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. doi: 10.1070/SM1999v190n03ABEH000395. Google Scholar

[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. doi: 10.1023/A:1019273927768. Google Scholar

[16]

E. Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws,, J. Hyperbolic Differ. Equ., 2 (2005), 885. doi: 10.1142/S0219891605000658. Google Scholar

[17]

E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws,, J. Hyperbolic Differ. Equ., 4 (2007), 729. doi: 10.1142/S0219891607001343. Google Scholar

[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. doi: 10.1007/s00205-009-0217-x. Google Scholar

[19]

E. Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property,, J. Math. Sci., 159 (2009), 180. doi: 10.1007/s10958-009-9434-y. Google Scholar

[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. doi: 10.1137/080724587. Google Scholar

[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. doi: 10.1016/j.anihpc.2012.12.009. Google Scholar

[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,, , (). Google Scholar

[23]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Univ. Press, (1970). Google Scholar

[24]

L. Tartar, Compensated compactness and applications to partial differential equations,, in Nonlinear analysis and mechanics, 39 (1979), 136. Google Scholar

[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. doi: 10.1017/S0308210500020606. Google Scholar

show all references

References:
[1]

A. S. Besicovitch, Almost Periodic Functions,, Cambridge University Press, (1932). Google Scholar

[2]

G.-Q. Chen and H. Frid, Decay of entropy solutions of nonlinear conservation laws,, Arch. Rational Mech. Anal., 146 (1999), 95. doi: 10.1007/s002050050138. Google Scholar

[3]

G.-Q. Chen and Y.-G. Lu, The study on application way of the compensated compactness theory,, Chinese Sci. Bull., 34 (1989), 15. Google Scholar

[4]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, 3rd edition, (2010). doi: 10.1007/978-3-642-04048-1. Google Scholar

[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. doi: 10.1137/130909688. Google Scholar

[6]

R. J. DiPerna, Measure-valued solutions to conservation laws,, Arch. Rational Mech. Anal., 88 (1985), 223. doi: 10.1007/BF00752112. Google Scholar

[7]

P. Gerárd, Microlocal defect measures,, Comm. Partial Diff. Equat., 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar

[8]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups,, Providence, (1957). Google Scholar

[9]

S. N. Kruzhkov, First order quasilinear equations in several independent variables,, Mat. Sb., 81 (1970), 228. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media,, Comp. GeoSci., 14 (2010), 105. doi: 10.1007/s10596-009-9135-0. Google Scholar

[13]

E. Yu. Panov, On sequences of measure-valued solutions of first-order quasilinear equations,, Mat. Sb., 185 (1994), 87. doi: 10.1070/SM1995v081n01ABEH003621. Google Scholar

[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. doi: 10.1070/SM1999v190n03ABEH000395. Google Scholar

[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. doi: 10.1023/A:1019273927768. Google Scholar

[16]

E. Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws,, J. Hyperbolic Differ. Equ., 2 (2005), 885. doi: 10.1142/S0219891605000658. Google Scholar

[17]

E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws,, J. Hyperbolic Differ. Equ., 4 (2007), 729. doi: 10.1142/S0219891607001343. Google Scholar

[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. doi: 10.1007/s00205-009-0217-x. Google Scholar

[19]

E. Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property,, J. Math. Sci., 159 (2009), 180. doi: 10.1007/s10958-009-9434-y. Google Scholar

[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. doi: 10.1137/080724587. Google Scholar

[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. doi: 10.1016/j.anihpc.2012.12.009. Google Scholar

[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,, , (). Google Scholar

[23]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Univ. Press, (1970). Google Scholar

[24]

L. Tartar, Compensated compactness and applications to partial differential equations,, in Nonlinear analysis and mechanics, 39 (1979), 136. Google Scholar

[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. doi: 10.1017/S0308210500020606. Google Scholar

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