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Darko Mitrovic

Publications and source records attributed to Darko Mitrovic.

24 records · Page 2Linked to original sources
arXiv2012

Velocity averaging -- a general framework

We prove that the sequence of averaged quantities $\int_{\R^m}u_n(\mx,\msnop)$ $ρ(\msnop)d\msnop$, is strongly precompact in $\Ldl\Rd$, where $ρ\in \Ldc{\R^m}$, and $u_n\in \Ld{\R^m; \pL s\Rd}$, $s\geq 2$, are weak solutions to differential operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators. If $s>2$ then the coefficients can be discontinuous with respect to the space variable $\mx\in \R^d$, otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the $H$-measures.

math.AP↗
arXiv2012

Singular solutions of a fully nonlinear 2x2 system of conservation laws

Existence and admissibility of $δ$-shock type solution is discussed for the following nonconvex strictly hyperbolic system arising in studues of plasmas: \pa_t u + \pa_x \big(\Sfrac{u^2+v^2}{2} \big) &=0 \pa_t v +\pa_x(v(u-1))&=0. The system is fully nonlinear, i.e. it is nonlinear with respect to both variables. The latter system does not admit the classical Lax-admissible solution to certain Riemann problems. By introducing complex valued corrections in the framework of the weak asymptotic method, we show that an compressive $δ$-shock type solution resolves such Riemann problems. By letting the approximation parameter to zero, the corrections become real valued and we obtain a $δ$-type solution concept. In the frame of that concept, we can show that every $2\times 2$ system of conservation laws admits $δ$-type solution.

math.AP↗
arXiv2011

H-distributions - an extension of the H-measures

We use the continuity of Fourier multiplier operators on $L^p$ to introduce the $H$-distributions --- an extension of $H$-measures in the $L^p$ framework. We apply the $H$-distributions to obtain an $L^p$ version of the localisation principle, and reprove the $L^p$---$L^q$ variant of the Murat--Tartar div-curl lemma.

math.FA↗
arXiv2011

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

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 $L^1_{loc}$ precompactness of such quasi-solutions.

math.AP↗
arXiv2010

New entropy conditions for scalar conservation laws with discontinuous flux

We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under assumptions on the flux which provide the maximum principle. In particular, we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions.

math.AP↗