arXiv subjects

Maja Tasković

Publications and source records attributed to Maja Tasković.

12 recordsLinked to original sources

arXiv2025

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

We study a fuzzy variant of the inhomogeneous Landau equation and establish global-in-time existence and uniqueness of smooth solutions for moderately soft potentials. The spatial delocalization introduced in the collision operator not only enhances regularity and prevents singularity formation, 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.

math.AP↗

arXiv2025

Global regularity and decay estimates for the relativistic Landau equation

We consider the relativistic Landau equation in the spatially inhomogeneous, far-from-equilibrium regime. We establish regularity estimates of all orders, implying that solutions remain smooth for as long as some zeroth-order conditional bounds hold. We also prove that polynomial and exponential decay in the momentum variable is propagated forward in time. As part of our proof, we establish a Schauder estimate for linear relativistic kinetic equations, that may be of independent interest.

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

Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation

We present in this document the Lebesgue and Sobolev propagation of exponential tails for solutions of the homogeneous Boltzmann equation for hard and Maxwell interactions. In addition, we show the $L^{p}$-integrability creation of such tails in the case of hard interactions. The document also presents a result on exponentially-fast convergence to thermodynamical equilibrium and propagation of singularities and regularization of such solutions. All these results are valid under the mere Grad's cut-off condition for the angular scattering kernel. Highlights of this contribution include: (1) full range of $L^{p}$-norms with $p\in[1,\infty]$, (2) analysis for the critical case of Maxwell interactions, (3) propagation of fractional Sobolev exponential tails using pointwise conmutators, and (4) time asymptotic and propagation of regularity and singularities under general physical data. In many ways, this work is an improvement and an extension of several classical works in the area; we use known techniques and introduce new and flexible ideas that achieve the proofs in an elementary manner.

math-ph↗

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.

math.AP↗

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.

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.

math.AP↗

arXiv2018

On pointwise exponentially weighted estimates for the Boltzmann equation

In this paper we prove propagation in time of weighted $L^\infty$ bounds for solutions to the non-cutoff homogeneous Boltzmann equation that satisfy propagation in time of weighted $L^1$ bounds. To emphasize that the propagation in time of weighted $L^{\infty}$ bounds relies on the propagation in time of weighted $L^1$ bounds, we express our main result using certain general weights. Consequently we apply the main result to cases of exponential and Mittag-Leffler weights, for which propagation in time of weighted $L^1$ bounds holds.

math.AP↗

arXiv2018

Entropy dissipation estimates for the relativistic Landau equation, and applications

In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.

math.AP↗

arXiv2017

Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

We study the spatially homogeneous Boltzmann equation for Maxwell molecules, and its $1$-dimensional model, the Kac equation. We prove propagation in time of stretched exponential moments of their weak solutions, both for the angular cutoff and the angular non-cutoff case. The order of the stretched exponential moments in question depends on the singularity rate of the angular kernel of the Boltzmann and the Kac equation. One of the main tools we use are Mittag-Leffler moments, which generalize the exponential ones.

math.AP↗

arXiv2017

On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff

We establish the $L^1$ weighted propagation properties for solutions of the Boltzmann equation with hard potentials and non-integrable angular components in the collision kernel. Our method identifies null forms by angular averaging and deploys moment estimates of solutions to the Boltzmann equation whose summability is achieved by introducing the new concept of Mittag-Leffler moments - extensions of $L^1$ exponentially weighted norms. Such $L^1$ weighted norms of solutions to the Boltzmann equation are, both, generated and propagated in time and the characterization of their corresponding Mittag-Leffler weights depends on the angular singularity and potential rates in the collision kernel. These estimates are a fundamental step in order to obtain $L^\infty$ exponentially weighted estimates for solutions of the Boltzmann equation being developed in a follow up work.

math.AP↗

arXiv2016

On dispersive blow-ups for the nonlinear Schrödinger equation

In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schrödinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow up on a straight line and on a sphere.

math.AP↗