A generalization of H-measures and application on purely fractional scalar conservation laws

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November 2011, 10(6): 1617-1627. doi: 10.3934/cpaa.2011.10.1617

A generalization of $H$-measures and application on purely fractional scalar conservation laws

Darko Mitrovic ^1, and Ivan Ivec ^2,

1.	Faculty of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica
2.	Faculty of Mathematics, University of Zagreb, Bijenicka cesta 30, 10000 Zagreb, Croatia

Received January 2010 Revised March 2011 Published May 2011

Abstract
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We extend the notion of $H$-measures on test functions defined on $R^d\times P$, where $P\subset R^d$ is an arbitrary compact simply connected Lipschitz manifold such that there exists a family of regular nonintersecting curves issuing from the manifold and fibrating $R^d$. We introduce a concept of quasi-solutions to purely fractional scalar conservation laws and apply our extension of the $H$-measures to prove strong $L_{l o c}^1$ precompactness of such quasi-solutions.

Keywords: $H$-measures, fractional conservation laws..

Mathematics Subject Classification: Primary: 35L99, 35L65; Secondary: 42B1.

Citation: Darko Mitrovic, Ivan Ivec. A generalization of $H$-measures and application on purely fractional scalar conservation laws. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1617-1627. doi: 10.3934/cpaa.2011.10.1617

References:

[1]	J. Aleksić, D. Mitrovíc and S. Pilipović, Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media, Journal of Evolution Equations, 9 (2009), 809-828. doi: doi:10.1007/s00028-009-0035-5.
[2]	N. Alibaud, Entropy formulation for fractal conservation laws, Journal of Evolution Equations, 7 (2007), 145-175. doi: doi:10.1007/s00028-006-0253-z.
[3]	N. Antonic and M. Lazar, Parabolic variant of H-measures in homogenisation of a model problem based on Navier-Stokes equation, Nonlinear Analysis-Real World Appl, 11 (2010), 4500-4512. doi: doi:10.1016/j.nonrwa.2008.07.010.
[4]	N. Antonic and M. Lazar, $H$-measures and variants applied to parabolic equations, J. Math. Anal. Appl., 343 (2008), 207-225. doi: doi:10.1016/j.jmaa.2007.12.077.
[5]	R. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292 (1985), 383-419. doi: doi:10.1090/S0002-9947-1985-0808729-4.
[6]	J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: doi:10.1007/s00205-006-0429-2.
[7]	P. Gerard, Microlocal Defect Measures, Comm. Partial Differential Equations, 16 (1991), 1761-1794. doi: doi:10.1080/03605309108820822.
[8]	S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sbornik., 81 (1970), 228-255; English transl. in Math. USSR Sb., 10 (1970), 217-243.
[9]	P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. of American Math. Soc., 7 (1994), 169-191. doi: doi:10.1090/S0894-0347-1994-1201239-3.
[10]	P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, Part I. Ann. Inst. H. Poincare Sect. A (N.S.), 1 (1984), 109-145, 223-283.
[11]	P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, Part II. Ann. Inst. H.Poincare Sect. A (N.S.), 1 (1984), 109-145, 223-283.
[12]	D. Mitrovic, Existence and stability of a multidimensional scalar conservation law with discontinuous flux, Netw. Het. Media, 5 (2010), 163-188. doi: doi:10.3934/nhm.2010.5.163.
[13]	E. Yu. Panov, On sequences of measure-valued solutions of a first order quasilinear equations, Russian Acad. Sci. Sb. Math., 81 (1995), 211-227. doi: doi:10.1070/SM1995v081n01ABEH003621.
[14]	E. Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong pre-compactness property, Journal of Mathematical Sciences, 159 (2009), 180-228. doi: doi:10.1007/s10958-009-9434-y.
[15]	E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, Journal of Hyperbolic Differential Equations, 4 (2007), 729-770. doi: doi:10.1142/S0219891607001343.
[16]	E. Yu. Panov, Existence and strong precompactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 195 (2010), 643-673. doi: doi:10.1007/s00205-009-0217-x.
[17]	S. A. Sazhenkov, The genuinely nonlinear Graetz-Nusselt ultraparabolic equation, (Russian. Russian summary) Sibirsk. Mat. Zh., 47 (2006), 431-454; translation in Siberian Math. J., 47 (2006), 355-375. doi: doi:10.1007/s11202-006-0048-z.
[18]	L. Tartar, H-measures, a new approach for studying homogenisation, oscillation and concentration effects in PDEs, Proc. Roy. Soc. Edinburgh. Sect. A, 115 (1990), 193-230.
[19]	L. Tartar, "The General Theory of Homogenization. A Personalized Introduction," Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009. xxii+470 pp.

show all references

References:

[1]	J. Aleksić, D. Mitrovíc and S. Pilipović, Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media, Journal of Evolution Equations, 9 (2009), 809-828. doi: doi:10.1007/s00028-009-0035-5.
[2]	N. Alibaud, Entropy formulation for fractal conservation laws, Journal of Evolution Equations, 7 (2007), 145-175. doi: doi:10.1007/s00028-006-0253-z.
[3]	N. Antonic and M. Lazar, Parabolic variant of H-measures in homogenisation of a model problem based on Navier-Stokes equation, Nonlinear Analysis-Real World Appl, 11 (2010), 4500-4512. doi: doi:10.1016/j.nonrwa.2008.07.010.
[4]	N. Antonic and M. Lazar, $H$-measures and variants applied to parabolic equations, J. Math. Anal. Appl., 343 (2008), 207-225. doi: doi:10.1016/j.jmaa.2007.12.077.
[5]	R. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc., 292 (1985), 383-419. doi: doi:10.1090/S0002-9947-1985-0808729-4.
[6]	J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: doi:10.1007/s00205-006-0429-2.
[7]	P. Gerard, Microlocal Defect Measures, Comm. Partial Differential Equations, 16 (1991), 1761-1794. doi: doi:10.1080/03605309108820822.
[8]	S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sbornik., 81 (1970), 228-255; English transl. in Math. USSR Sb., 10 (1970), 217-243.
[9]	P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. of American Math. Soc., 7 (1994), 169-191. doi: doi:10.1090/S0894-0347-1994-1201239-3.
[10]	P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, Part I. Ann. Inst. H. Poincare Sect. A (N.S.), 1 (1984), 109-145, 223-283.
[11]	P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, Part II. Ann. Inst. H.Poincare Sect. A (N.S.), 1 (1984), 109-145, 223-283.
[12]	D. Mitrovic, Existence and stability of a multidimensional scalar conservation law with discontinuous flux, Netw. Het. Media, 5 (2010), 163-188. doi: doi:10.3934/nhm.2010.5.163.
[13]	E. Yu. Panov, On sequences of measure-valued solutions of a first order quasilinear equations, Russian Acad. Sci. Sb. Math., 81 (1995), 211-227. doi: doi:10.1070/SM1995v081n01ABEH003621.
[14]	E. Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong pre-compactness property, Journal of Mathematical Sciences, 159 (2009), 180-228. doi: doi:10.1007/s10958-009-9434-y.
[15]	E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, Journal of Hyperbolic Differential Equations, 4 (2007), 729-770. doi: doi:10.1142/S0219891607001343.
[16]	E. Yu. Panov, Existence and strong precompactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 195 (2010), 643-673. doi: doi:10.1007/s00205-009-0217-x.
[17]	S. A. Sazhenkov, The genuinely nonlinear Graetz-Nusselt ultraparabolic equation, (Russian. Russian summary) Sibirsk. Mat. Zh., 47 (2006), 431-454; translation in Siberian Math. J., 47 (2006), 355-375. doi: doi:10.1007/s11202-006-0048-z.
[18]	L. Tartar, H-measures, a new approach for studying homogenisation, oscillation and concentration effects in PDEs, Proc. Roy. Soc. Edinburgh. Sect. A, 115 (1990), 193-230.
[19]	L. Tartar, "The General Theory of Homogenization. A Personalized Introduction," Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009. xxii+470 pp.

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