# American Institute of Mathematical Sciences

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

 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.
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:
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##### 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. 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. 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. 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. 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. 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. doi: doi:10.1007/s00205-006-0429-2. Google Scholar [7] P. Gerard, Microlocal Defect Measures,, Comm. Partial Differential Equations, 16 (1991), 1761. doi: doi:10.1080/03605309108820822. Google Scholar [8] S. N. Kruzhkov, First order quasilinear equations in several independent variables,, Mat. Sbornik., 81 (1970), 228. 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. 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. 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. Google Scholar [12] D. Mitrovic, Existence and stability of a multidimensional scalar conservation law with discontinuous flux,, Netw. Het. Media, 5 (2010), 163. 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. 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. 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. 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. 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. 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. Google Scholar [19] L. Tartar, "The General Theory of Homogenization. A Personalized Introduction,", Lecture Notes of the Unione Matematica Italiana, (2009). Google Scholar
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