arXiv subjects

Darko Mitrovic

Publications and source records attributed to Darko Mitrovic.

At least 19 recordsLinked to original sources

arXiv2026

Existence of strong initial traces for stochastic conservation laws

We prove existence and uniqueness of a strong initial trace for every bounded kinetic solution of a stochastic scalar conservation law, although no initial value is prescribed and no nondegeneracy condition is imposed on the flux. The core of the proof is pathwise. After fixing a realization, the rescaled stochastic terms and kinetic measure vanish in the blow-up limit, so Panov's compactness argument applies; degenerate flux intervals are handled by recursive dimension reduction. The stochastic setting creates several additional difficulties. The martingale identities must remain valid on one common full-probability set throughout the reductions, which requires a parameterized stochastic-Fubini construction. Moreover, the pathwise trace is not automatically measurable because its exceptional sets may depend on the realization. Deterministic time averages and right-continuity of the filtration yield a jointly measurable initial trace, with local strong convergence along essential times, both almost surely and in mean.

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arXiv2026

Well-posedness and a sign-graph limit for an active Cahn--Hilliard equation on Riemannian manifolds

We establish a weak solution theory and a singular constitutive limit for a non variational Cahn Hilliard evolution on compact Riemannian manifolds, possibly with boundary. The model combines the classical double well chemical potential with an active correction concentrated in diffuse transition layers and sensitive to the sign of the Laplace Beltrami operator, together with a monotone anchoring mechanism toward a prescribed reference state. The active correction breaks the passive gradient flow structure and introduces a nonlinear dependence on second derivatives not identified by the natural weak compactness estimates. For square integrable initial data and sufficiently integrable reference data, we construct weak solutions in arbitrary dimension. A Galerkin level one sided comparison argument, using monotonicity of the classifier and anchoring law together with biharmonic coercivity, yields strong convergence of the approximate Laplacians and hence strong second order compactness. This identifies the nonlinear active term in an ordinary weak formulation, without a second derivative defect. In two dimensions we prove uniqueness and continuous dependence. The estimates depend only on the bound and monotonicity of the classifier, not on its slope, and are therefore uniform for increasingly steep arctangent classifiers. Their singular limit is governed by the maximal monotone sign graph acting on the Laplace Beltrami operator. In arbitrary dimension we obtain subsequential strong second order convergence to a weak solution of the resulting differential inclusion. In two dimensions the limiting state is unique, so the entire steep classifier family converges; uniqueness of the constitutive multiplier on the zero Laplacian set is not asserted. This is a constitutive steepness limit at fixed diffuse interface thickness, rather than a sharp interface limit.

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arXiv2026

Multidimensional scalar conservation laws with discontinuous flux: well-posedness without non-degeneracy

We establish existence, uniqueness, and local $L^1$-stability for multidimensional scalar conservation laws with discontinuous heterogeneous flux, without imposing a non-degeneracy condition on the physical space--time flux. The discontinuity set may be a locally finite family of $C^2$ hypersurfaces whose intersections and singular points are contained in a closed set of vanishing $(d-1)$-dimensional Hausdorff measure. The main obstacle is the possible failure of multidimensional compactness when a nontrivial linear combination of the space--time flux components is constant on an interval of states. We show that, for interface problems, this obstruction can be reduced to the physical normal flux. Its flat intervals are collapsed by a normal-flux quotient that preserves the normal flux and hence the Rankine--Hugoniot relation. On the non-flat region, the time and tangential components are replaced by auxiliary polynomial functions of the normal flux, while the physical normal component is left unchanged. The resulting auxiliary space--time vector satisfies the interval non-degeneracy required by Panov's compactness theory. This yields strong compactness and strong one-sided traces of the quotient variables. The interface admissibility condition is obtained by projecting the vanishing-viscosity germ associated with the two one-sided physical normal fluxes. A localized viscous comparison shows that the quotient traces belong to this maximal $L^1$-dissipative germ, and a Young-measure contraction argument recovers strong convergence of the physical states. Curved interfaces are treated by flattening, localization, and finite-propagation patching. The theory is first constructed for $BV$ initial data and then extended by continuity to $L^1_{\mathrm{loc}}\cap L^\infty$ data taking values in the invariant interval.

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arXiv2026

A high-frequency tail condition and a diagnostic iteration for the Navier--Stokes equations

We consider Leray solutions of the three--dimensional incompressible Navier--Stokes equations on $\R^3$ with smooth, rapidly decaying initial data. The analysis is based on a frequency decomposition into low and high modes via the cutoffs $\A_R=ϕ(|D|/R)$ and $\A^R=I-\A_R$. Combining the energy inequality with Bernstein estimates yields uniform control of the low--frequency component $\A_R\u$. For the high--frequency component we assume a quantitative \emph{turbulence condition}, requiring that the solution possesses a non--negligible high--frequency tail in $L^\infty$ (in fact, it suffices to impose this condition only on a terminal time layer near a putative blow--up time). Under this hypothesis we introduce a time--localized diagnostic Picard iteration adapted to $\A^R\u$. Using a uniform $L^\infty$ estimate of Giga--Inui--Matsui type (with the cutoff $\A^R$) together with high--frequency heat--flow decay, we show that the iteration is contractive and converges to $\A^R\u$, providing a uniform bound for $\A^R\u$ up to the maximal time of boundedness. Consequently, the turbulence regime is incompatible with finite--time blow--up: any Leray solution satisfying the turbulence condition is bounded, and hence smooth, for all times (equivalently, it cannot blow up in finite time).

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arXiv2025

Scalar conservation laws with discontinuous flux: existence and uniqueness

We study the well-posedness of the Cauchy problem for scalar conservation laws with discontinuous, non-degenerate fluxes. Locally, the fluxes are piecewise smooth across interfaces described by a Heaviside-type discontinuity, with left and right states depending smoothly on both space and the solution variable. The interface is given by a smooth function, and the fluxes vanish at the boundary values of the admissible interval for the solution. In addition, we consider the more general case of heterogeneous flux functions with bounded variation in the spatial variable and smooth dependence on the solution variable, again vanishing at the prescribed boundary states. For this setting, we construct a stable semigroup of solutions, thereby establishing a well-posed solution framework.

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arXiv2024

Navigating the Complex Landscape of Shock Filter Cahn-Hilliard Equation: From Regularized to Entropy Solutions

Image inpainting involves filling in damaged or missing regions of an image by utilizing information from the surrounding areas. In this paper, we investigate a highly nonlinear partial differential equation inspired by the modified Cahn-Hilliard equation. Instead of using standard potentials that depend solely on pixel intensities, we consider morphological image enhancement filters that are based on a variant of the shock filter: : \begin{align*} \partial_t u &= Δ\left(-ν\arctan(Δu)|\nabla u| - μΔu \right)+ λ(u_0 - u). \end{align*} This is referred to as the Shock Filter Cahn-Hilliard Equation. This equation is nonlinear with respect to the second-order derivative, which poses significant mathematical challenges. To address these, we make use of a specific approximation argument, establishing the existence of a family of approximate solutions through the Leray-Schauder fixed point theorem and the Aubin-Lions lemma. In the limit, we obtain a solution strategy wherein we can prove the existence and uniqueness of solutions. Proving the latter involves the use of Young measures and Kruzhkov entropy type-admissibility conditions. Additionally, we use a numerical method based on the convexity splitting idea to approximate solutions of the nonlinear partial differential equation and achieve fast inpainting results. To demonstrate the effectiveness of our approach, we apply our method to standard binary images and compare it with variations of the Cahn-Hilliard equation commonly used in the field.

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arXiv2024

Pre-electoral coalition agreement from the Black-Scholes point of view

A political party can be considered as a company whose value depends on the voters support i.e. on the percentage of population supporting the party. Dynamics of the support is thus as a stochastic process with a deterministic growth rate perturbed by a white noise modeled through the Wiener process. This is in an analogy with the option modeling where the stock price behaves similarly as the voters' support. While in the option theory we have the question of fair price of an option, the question that we ask here is what is a reasonable level of support that the coalition of a major party (safely above the election threshold) and a minor party (under or around the election threshold) should achieve in order the minor party to get one more representative. We shall elaborate some of the conclusions in the case of recent elections in Montenegro (June, 2023) which are particularly interesting due to lots of political subjects entering the race.

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arXiv2023

A dynamic capillarity equation with stochastic forcing on manifolds: a singular limit problem

We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold $(M,g)$. \begin{equation*}\tag{P} d \left(u_{\varepsilon,δ}-δΔ u_{\varepsilon,δ}\right) +\operatorname{div} f_{\varepsilon}(x, u_{\varepsilon,δ})\, dt =\varepsilon Δu_{\varepsilon,δ}\, dt Φ(x, u_{\varepsilon,δ})\, dW_t, \end{equation*} where $f_{\varepsilon}$ is a sequence of smooth vector fields converging in $L^p(M\times \Bbb{R})$ ($p>2$) as $\varepsilon\downarrow 0$ towards a vector field $f\in L^p(M;C^1(\Bbb{R}))$, and $W_t$ is a Wiener process defined on a filtered probability space. First, for fixed values of $\varepsilon$ and $δ$, we establish the existence and uniqueness of weak solutions to the Cauchy problem for (P). Assuming that $f$ is non-degenerate and that $\varepsilon$ and $δ$ tend to zero with $δ/\varepsilon^2$ bounded, we show that there exists a subsequence of solutions that strongly converges in $L^1_{ω,t,x}$ to a martingale solution of the following stochastic conservation law with discontinuous flux: $$ d u +\operatorname{div} f(x, u)\,dt=Φ(u)\, dW_t. $$ The proofs make use of Galerkin approximations, kinetic formulations as well as $H$-measures and new velocity averaging results for stochastic continuity equations. The analysis relies in an essential way on the use of a.s.~representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.

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arXiv2022

Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

We consider the degenerate parabolic equation $$ \partial_t u +\mathrm{div} {\mathfrak f}_{\bf x}(u)=\mathrm{div}(\mathrm{div} ( A_{\bf x}(u) ) ), \ \ {\bf x} \in M, \ \ t\geq 0 $$ on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\in {\mathbb R}$, ${\bf x}\mapsto {\mathfrak f}_{\bf x}(u)$ is a vector field and ${\bf x}\mapsto A_{\bf x}(u)$ is a $(1,1)$-tensor field on $M$ such that $u\mapsto \langle A_{\bf x}(u) {\boldsymbol ξ},{\boldsymbol ξ} \rangle$, ${\boldsymbol ξ}\in T_{\bf x} M$, is non-decreasing with respect to $u$. The fact that the notion of divergence appearing in the equation depends on the metric $g$ requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem.

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arXiv2020

A vanishing dynamic capillarity limit equation with discontinuous flux

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the equation \begin{equation*} \begin{cases} \partial_t u_{\varepsilon,δ} +\mathrm{div} {\mathfrak f}_{\varepsilon,δ}({\bf x}, u_{\varepsilon,δ})=\varepsilon Δu_{\varepsilon,δ}+δ(\varepsilon) \partial_t Δu_{\varepsilon,δ}, \ \ {\bf x} \in M, \ \ t\geq 0 u|_{t=0}=u_0({\bf x}). \end{cases} \end{equation*} Here, ${\mathfrak f}_{\varepsilon,δ}$ and $u_0$ are smooth functions while $\varepsilon$ and $δ=δ(\varepsilon)$ are fixed constants. Assuming ${\mathfrak f}_{\varepsilon,δ} \to {\mathfrak f} \in L^p( \mathbb{R}^d\times \mathbb{R};\mathbb{R}^d)$ for some $1<p<\infty$, strongly as $\varepsilon\to 0$, we prove that, under an appropriate relationship between $\varepsilon$ and $δ(\varepsilon)$ depending on the regularity of the flux ${\mathfrak f}$, the sequence of solutions $(u_{\varepsilon,δ})$ strongly converges in $L^1_{loc}(\mathbb{R}^+\times \mathbb{R}^d)$ towards a solution to the conservation law $$ \partial_t u +\mathrm{div} {\mathfrak f}({\bf x}, u)=0. $$ The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second.

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arXiv2018

Well-posedness for stochastic scalar conservation laws on Riemannian manifolds

We consider the scalar conservation law with stochastic forcing $$ \partial_t u +\mathrm{div}_g {\mathfrak f}(\mx,u)= Φ(\mx,u) dW, \ \ {\bf x} \in M, \ \ t\geq 0 $$ on a smooth compact Riemannian manifold $(M,g)$ where $W$ is the Wiener process and ${\bf x}\mapsto {\mathfrak f}(\mx,ξ)$ is a vector field on $M$ for each $ξ\in {\bf R}$. We introduce admissibility conditions, derive the kinetic formulation and use it to prove well posedness.

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arXiv2018

Existence and Uniqueness of Singular Solutions for a Conservation Law Arising in Magnetohydrodynamics

The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). The system has the form \begin{align*} \partial_t u+\partial_x \Big({\textstyle \frac{u^2+v^2}{2}}\Big)=0,\\ \partial_t v+\partial_x \big(v(u-1)\big)=0. \end{align*} It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set $v=0$. As a consequence, certain Riemann problems have no weak solutions in the traditional class of functions of bounded variation. It was argued in Nonlinearity 9, 1547--1563 (1996) that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Solutions of this type were exhibited in Proc. Edinb. Math. Soc. 55, 711--729 (2012) and Russ. J. Math. Phys. 22, 518--527 (2015), but uniqueness was not obtained. In the current work, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and it can be shown that admissible solutions are unique in the framework developed here.

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arXiv2017

The structure of ${\cal A}$-free measures with uniformly singular part

We prove that a singular part $μ_s$ of a measure $μ$ satisfying ${\cal A}μ=0$ for a linear partial differential operator ${\cal A}$ defined on $R^d$ has the range in the intersection of kernels of the principal symbol of ${\cal A}$ if the singular part is singular with respect to all the variables (uniformly singular) i.e. it is such that for $μ_s$-almost every $x\in R^d$ there exist positive functions $α(ε), β(ε)$, $ε\in R$, satisfying $\frac{α(ε)}ε\to 0$, $ \fracε{β(ε)}\to 0$ and a set $E_ε\subset B(\mx,α(ε))$ such that $\lim_{ε\to 0}\frac{μ_s(B(x,β(ε)) / E_ε)}{|μ_s|(E_ε)}=0$.

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arXiv2015

Transport-collapse scheme for scalar conservation laws -- initial and boundary value problems

We extend Brenier's transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws and initial-boundary value problem for homogeneous scalar conservation laws. It is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. In the case of initial-boundary value problem, we such a procedure is used to construct a numerical scheme which leads us to a new solution concept for initial-boundary value problem for scalar conservation laws. The concept is a generalization (refinement) of the previous works on initial-boundary value problem. We also provide numerical examples.

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arXiv2014

On a generalization of compensated compactness in the $L^p-L^q$ setting

We investigate conditions under which, for two sequences $(u_r)$ and $(v_r)$ weakly converging to $u$ and $v$ in $L^p(R^d;R^N)$ and $L^{q}(R^d;R^N)$, respectively, $1/p+1/q \leq 1$, a quadratic form $q(x;u_r,v_r)=\sum\limits_{j,m=1}^N q_{j m}(x)u_{j r} v_{m r}$ converges toward $q(x;u,v)$ in the sense of distributions. The conditions involve fractional derivatives and variable coefficients, and they represent a generalization of the known compensated compactness theory. The proofs are accomplished using a recently introduced $H$-distribution concept. We apply the developed techniques to a nonlinear (degenerate) parabolic equation.

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arXiv2014

Entropy conditions for scalar conservation laws with discontinuous flux revisited

We propose new entropy admissibility conditions for multidimensional hyperbolic scalar conservation laws with discontinuous flux which generalize one-dimensional Karlsen-Risebro-Towers entropy conditions. These new conditions are designed, in particular, in order to characterize the limit of vanishing viscosity approximations. On the one hand, they comply quite naturally with a certain class of physical and numerical modeling assumptions; on the other hand, their mathematical assessment turns out to be intricate. \smallskip The generalization we propose is not only with respect to the space dimension, but mainly in the sense that the "crossing condition" of [K.H. Karlsen, N.H. Risebro, J. Towers, Skr.\,K.\,Nor.\,Vid.\,Selsk. (2003)] is not mandatory for proving uniqueness with the new definition. We prove uniqueness of solutions and give tools to justify their existence via the vanishing viscosity method, for the multi-dimensional spatially inhomogeneous case with a finite number of Lipschitz regular hypersurfaces of discontinuity for the flux function.

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arXiv2014

On the velocity averaging for equations with optimal heterogeneous rough coefficients

Assume that $(u_n)$ is a sequence of solutions to heterogeneous equations with rough coefficients and fractional derivatives, weakly converging to zero in ${\rm L}^p(\R^{d+m})$, with $p>1$. We prove that the sequence of averaged quantities $(\int ρ(\my) u_n(\mx,\my) d\my)$ is strongly precompact in $\Ljl\Rd$ for any $ρ\in \Cc{\R^m}$, provided that restrictive non-degeneracy conditions are satisfied. These are fulfilled for elliptic, parabolic, fractional convection-diffusion equations, as well as for parabolic equations with a fractional time derivative. The main tool that we are using is an adapted version of H-distributions. As a consequence of the introduced methods, we obtain an optimal velocity averaging result in the $\LL p$, $p\geq 2$, framework under the standard non-degeneracy conditions, as well as a connection between the H-measures and the H-distributions.

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arXiv2013

Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations

We prove that if traceability conditions are fulfilled then a weak solution $h\in L^\infty(\R^+\times\R^d\times \R)$ to {the ultra-parabolic transport equation} \begin{equation*} \pa_t h + \Div_x \left(F(t,x,λ)h\right)=\sum\limits_{i,j=1}^k\pa^2_{x_i x_j}\left(b_{ij}(t,x,λ) h\right)+\pa_λγ(t,x,λ), \end{equation*} is such that for every $ρ\in C^1_c(\R)$, the velocity averaged quantity $\int_{\R}h(t,x,λ)$ $ρ(λ)dλ$ admits the strong $L^1_{\rm loc}(\R^d)$-limit as $t\to 0$, i.e. there exist $h_0(x,λ)\in L^1_{\rm loc}(\R^d\times \R)$ and the set $E\subset\R^+$ of full measure such that for every $ρ\in C^1_c(\R)$, $$ L^1_{\rm loc}(\R^d)-\lim\limits_{t\to 0, \; t\in E} \int_{\R} h(t,x,λ)ρ(λ)dλ= \int_{\R} h_0(x,λ) ρ(λ)dλ. $$ As a corollary, under the traceability conditions, we prove existence of strong traces for entropy solutions to ultraparabolic equations in heterogeneous media.

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