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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. Gerard, Microlocal Defect Measures, Comm. Partial Differential Equations, 16 (1991), 1761-1794. doi: doi:10.1080/03605309108820822.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. Gerard, Microlocal Defect Measures, Comm. Partial Differential Equations, 16 (1991), 1761-1794. doi: doi:10.1080/03605309108820822.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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