arXiv subjects

Nataša Pavlović

Publications and source records attributed to Nataša Pavlović.

At least 19 recordsLinked to original sources

arXiv2026

The fuzzy Landau equation: global well-posedness and Fisher information

We investigate a fuzzy variant of the inhomogeneous Landau equation and establish global-in-time existence and uniqueness of smooth solutions for a range of soft potentials. The spatial delocalization introduced in the collision operator not only prevents singularity formation and enhances regularity, but also reveals additional structural properties of the model. In particular, we show that several forms of the Fisher information decay monotonically or remain uniformly bounded in time.

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arXiv2026

Complex-valued modified Zakharov Kuznetsov equation

Motivated by the introduction of the Zakharov-Kuznetsov equation as a higher dimensional generalization of the Korteweg-de Vries equation, in this paper we introduce the modified Zakharov-Kuznetsov (mZK) equation as a 2-dimensional generalization of the complex-valued modified Korteweg-de Vries equation. We initiate the mathematical analysis of the mZK equation on $\mathbb{T}^2$ by proving a local well-posedness result in Sobolev spaces and by establishing a failure of uniform continuity. We also note that the real-valued version of the mZK equation has physical significance.

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arXiv2026

Low-regularity invariant measure for the complex-valued mKdV

In this paper we consider the twice-renormalized, complex-valued modified KdV (mKdV) on the one-dimensional torus introduced by Chapouto. Our main result is the construction of an invariant measure supported at low-regularity. This work complements the work of Kenig et al., which constructed invariant measures supported in higher-regularity spaces for the non-renormalized mKdV. Due to the low-regularity of the support of the measure, we are forced to work in Fourier-Lebesgue spaces. The fact that we consider the complex-valued mKdV makes the problem more complicated than the real-valued case, which was previously considered.

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arXiv2025

Emergence of fermion-mediated interactions in Bose-Fermi mixtures

This work is inspired by recent experimental observations in ultracold atomic Bose-Fermi mixtures [DeSalvo et al., Nature 568 (2019)]. These experiments reveal the emergence of an attractive fermion-mediated interaction between bosons, as well as a stability-instability transition. We give the first mathematical demonstration of this transition by studying the low-energy spectrum of a many-body interspecies Hamiltonian. More precisely, we show the convergence of its eigenvalues towards those of an effective Bose Hamiltonian, which includes fermion-mediated effects. Applying this result to a model with short-range potentials, we derive a stability-instability transition in the bosonic subsystem, driven by the Bose-Fermi coupling strength $g$. For small $|g|$, the bosons form a stable Bose-Einstein condensate with the energy per particle uniformly bounded from below. For large $|g|$, the energy per particle is no longer uniformly bounded from below, signaling the collapse of the condensate.

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arXiv2025

From the quantum Boltzmann operator to the quantum Landau operator

In this manuscript we derive the quantum Landau operator as the weak-coupling limit of the quantum Boltzmann operator (also known as the Uehling-Uhlenbeck operator). We consider both Fermi-Dirac and Bose-Einstein statistics. Our approach is inspired by the work by Benedetto and Pulvirenti, where the classical Landau operator was derived from the quantum Boltzmann operator. To capture the ternary term in the quantum Landau operator, we introduce a new two-parameter scaling that preserves the quantum effects in the limit. Furthermore, we provide an explicit rate of convergence that depends on the regularity of the interaction potential.

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arXiv2025

On the effective dynamics of Bose-Fermi mixtures

In this work, we describe the dynamics of a Bose-Einstein condensate interacting with a degenerate Fermi gas, at zero temperature. First, we analyze the mean-field approximation of the many-body Schrödinger dynamics and prove emergence of a coupled Hartree-type system of equations. We obtain rigorous error control that yields a non-trivial scaling window in which the approximation is meaningful. Second, starting from this Hartree system, we identify a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical; this is one of the main contributions of the present article. In this regime, the bosons are much lighter and more numerous than the fermions. We then prove convergence to a coupled Vlasov-Hartee system of equations with an explicit convergence rate.

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arXiv2025

An inverse Problem for the cubic $α$-NLS in Sobolev spaces

In this work, we address an inverse problem for a defocusing cubic nonlinear Schrödinger (NLS) equation in dimensions $d\in\{1, 2,3\}$ in a range of Sobolev spaces $H^s(\mathbb{R}^d)$ by employing the method of approximate solutions. We recover a smooth, space-dependent and compactly supported function $α$ that controls the nonlinearity (and thus self-interaction strength) in a multiplicative fashion. To the best of our knowledge, this is the first work based on approximate solutions in Sobolev spaces that treats an inverse problem for the NLS and provides explicit recovery of $α$.

math.AP↗

arXiv2025

Well-posedness and invariant measures for complex valued modified KdV equation

We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported on Sobolev spaces with increasing regularity. The fact that we work with complex valued functions makes the analysis of the invariance much harder compared to the real valued case, that can be handled instead following the ideas used by Zhidkov [73].

math.AP↗

arXiv2025

Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces

Six-wave interactions are used for modeling various physical systems, including in optical wave turbulence [16] (where a cascade of photons displays this kind of behavior) and in quantum wave turbulence [31] (for the interaction of Kelvin waves in superfluids). In this paper, we initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation. More precisely, we obtain the existence and uniqueness of non-negative mild solutions to this equation in exponentially weighted $L^\infty_{xv}$ spaces. This is accompanied by an analysis of the long-time behavior of such solutions - we prove that the solutions scatter, that is, they converge to solutions of the transport equation in the limit as $t \to \pm \infty$. Compared with the study of four-wave kinetic equations, the main challenge we face is to address the increased complexity of the geometry of the six-wave interactions.

math.AP↗

arXiv2024

Derivation of the Higher Order Boltzmann Equation for Hard Spheres

In this paper we complete the program initiated by the first and second authors and rigorously derive a Boltzmann-type equation that incorporates higher order collisions among gas particles. More precisely, starting from a finite $N$-particle system where the particles can perform symmetric hard sphere type interactions up to arbitrarily high order $M$, we derive a kinetic equation which consists of a linear combination of higher order collisional terms. We identify the new scaling regime that facilitates such collisions and systematically generalize geometric techniques allowing us to analyze the correlation of potentially distinct order recollisions as time evolves.

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arXiv2024

Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces

Inspired by ideas stemming from the analysis of the Boltzmann equation, in this paper we expand well-posedness theory of the spatially inhomogeneous 4-wave kinetic equation, and also analyze an infinite hierarchy of PDE associated with this nonlinear equation. More precisely, we show global in time well-posedness of the spatially inhomogeneous 4-wave kinetic equation for polynomially decaying initial data. For the associated infinite hierarchy, we construct global in time solutions using the solutions of the wave kinetic equation and the Hewitt-Savage theorem. Uniqueness of these solutions is proved by using a combinatorial board game argument tailored to this context, which allows us to control the factorial growth of the Dyson series.

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arXiv2024

On the global in time existence and uniqueness of solutions to the Boltzmann hierarchy

In this paper we establish the global in time existence and uniqueness of solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for the rigorous derivation of the Boltzmann equation from many particles. Inspired by available $L^{\infty}$-based a-priori estimate for solutions to the Boltzmann equation, we develop the polynomially weighted $L^\infty$ a-priori bounds for solutions to the Boltzmann hierarchy and handle the factorial growth of the number of terms in the Dyson's series by reorganizing the sum through a combinatorial technique known as the Klainerman-Machedon board game argument. This paper is the first work that exploits such a combinatorial technique in conjunction with an $L^{\infty}$-based estimate to prove uniqueness of the mild solutions to the Boltzmann hierarchy. Our proof of existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data is constructive and it employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.

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arXiv2022

Derivation of a Boltzmann equation with higher-order collisions from a generalized Kac model

In this work, we generalize M. Kac's original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order, and whose collision kernel is of Maxwell-type with cut-off.

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arXiv2022

On an $L^2$ critical Boltzmann equation

We prove the existence of a class of large global scattering solutions of Boltzmann's equation with constant collision kernel in two dimensions. These solutions are found for $L^2$ perturbations of an underlying initial data which is Gaussian jointly in space and velocity. Additionally, the perturbation is required to satisfy natural physical constraints for the total mass and second moments, corresponding to conserved or controlled quantities. The space $L^2$ is a scaling critical space for the equation under consideration. If the initial data is Schwartz then the solution is unique and again Schwartz on any bounded time interval.

math.AP↗

arXiv2022

Moment estimates and well-posedness of the binary-ternary Boltzmann equation

In this paper, we show generation and propagation of polynomial and exponential moments, as well as global well-posedness of the homogeneous binary-ternary Boltzmann equation. We also show that the co-existence of binary and ternary collisions yields better generation properties and time decay, than when only binary or ternary collisions are considered. To address these questions, we develop for the first time angular averaging estimates for ternary interactions. This is the first paper which discusses this type of questions for the binary-ternary Boltzmann equation and opens the door for studying moments properties of gases with higher collisional density.

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arXiv2022

A rigorous derivation of the Hamiltonian structure for the Vlasov equation

We consider the Vlasov equation in any spatial dimension, which has long been known to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie-Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to arXiv:1908.03847, which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison, and Weinstein on providing a "statistical basis" for the bracket structure of the Vlasov equation.

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arXiv2021

A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases

In this paper, we rigorously derive a Boltzmann equation for mixtures from the many body dynamics of two types of hard sphere gases. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases. As a corollary of the derivation, we prove Boltzmann's propagation of chaos assumption for the case of a mixtures of gases.

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arXiv2019

Higher order corrections to the mean-field description of the dynamics of interacting bosons

In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of $N$ $d$-dimensional bosons for large $N$. The bosons initially form a Bose-Einstein condensate and interact with each other via a pair potential of the form $(N-1)^{-1}N^{dβ}v(N^β\cdot)$ for $β\in[0,\frac{1}{4d})$. We derive a sequence of $N$-body functions which approximate the true many-body dynamics in $L^2(\mathbb{R}^{d N})$-norm to arbitrary precision in powers of $N^{-1}$. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.

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