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Andrej Zlatos

Publications and source records attributed to Andrej Zlatos.

At least 19 recordsLinked to original sources
arXiv2025

Well-Posedness for Low Regularity Solutions to the g-SQG Equation with Regular Level Sets

We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially H\"older continuous with H\"older exponents depending on the equation parameter $\alpha\in(0,\frac 12)$) that have $H^2$ level sets (i.e., with $L^2$ curvatures). Moreover, for $\alpha\le\frac 16$ and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose $H^2$-regularity, and hence not just due to level set collisions or "pile ups".

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arXiv2025

Maximal Double-Exponential Growth for the Euler Equation on the Half-Plane

We show that smooth solutions to the Euler equation on the half-plane can exhibit double-exponential growth of their vorticity gradients. We also determine the maximal possible growth rate and construct solutions that saturate it. These are the first such results on an unbounded resp. any 2D domain.

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arXiv2025

Well-Posedness and Finite Time Singularity for Touching g-SQG Patches on the Plane

We prove local well-posedness as well as singularity formation for the g-SQG patch model on the plane (so on a domain without a boundary), with $α\in(0,\frac 16]$ and patches being allowed to touch each other. We do this by bypassing any auxiliary contour equations and tracking patch boundary curves directly instead of their parametrizations. In our results, which are sharp in terms of $α$, the patch boundaries have $L^2$ curvatures and a singularity occurs when at least one of these $L^2$-norms blows up in finite time.

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arXiv2024

The 2D Muskat Problem I: Local Regularity on the Half-plane, Plane, and Strips

We prove local well-posedness for the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable layer (e.g., bedrock). Our result allows for the interface to touch the bottom, and hence applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer. We use this result in the companion paper [43] to prove existence of finite time stable regime singularities in this model, including for arbitrarily small initial data. We do not require the interface and its derivatives to vanish at $\pm\infty$ or be periodic, and even allow it to be $O(|x|^{1-})$, which is an optimal bound on the power of growth. We also extend our results to the cases of the Muskat problem on the whole plane and on horizontal strips, where almost all previous works did impose such limiting requirements.

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arXiv2024

The 2D Muskat Problem II: Stable Regime Small Data Singularity on the Half-plane

We study the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable layer (e.g., bedrock). Existence of finite time stable regime interface curve singularities is still open on the whole plane, but we show that they do arise on the half-plane, including from arbitrarily small smooth initial data. To obtain this result, we establish maximum principles for both the potential energy and the slope of solutions in this model, as well as develop a general local well-posedness theory in the companion paper [25].

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arXiv2023

Local regularity and finite time singularity for the generalized SQG equation on the half-plane

We show that the generalized SQG equation with $α\in(0,\frac 14]$ is locally well-posed on the half-plane in spaces of bounded integrable solutions that are natural for its dynamic on domains with boundaries, and allow for some power growth of the solution derivative in the normal direction at the boundary. We also show existence of solutions exhibiting finite time blow-up in the whole local well-posedness parameter regime $α\in(0,\frac 14]$, which is the first finite time singularity result for equations (as opposed to patch models) of this type. Moreover, we prove optimality of both these results by showing ill-posedness of the PDE in all the above spaces when $α>\frac 14$.

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arXiv2023

Optimal Estimates on the Propagation of Reactions with Fractional Diffusion

We study the reaction-fractional-diffusion equation $u_t+(-Δ)^{s} u=f(u)$ with ignition and monostable reactions $f$, and $s\in(0,1)$. We obtain the first optimal bounds on the propagation of front-like solutions in the cases where no traveling fronts exist. Our results cover most of these cases, and also apply to propagation from localized initial data.

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arXiv2022

Uniqueness of Positive Vorticity Solutions to the 2D Euler Equations on Singular Domains

We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$ boundaries and having corners with arbitrary angles, and can fail without the sign hypothesis when the domain has large angle corners. Hence positive vorticity solutions on such domains are Lagrangian, and we also obtain their uniqueness if the vorticity is initially constant near the boundary.

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arXiv2022

Virtual Linearity for KPP Reaction-Diffusion Equations

We show that long time solution dynamic for general reaction-advection-diffusion equations with KPP reactions is virtually linear in the following sense. Its leading order depends on the non-linear reaction only through its linearization at $u=0$, and it can also be recovered for general initial data by instead solving the PDE for restrictions of the initial condition to unit cubes on $\bbR^d$ (the latter means that non-linear interaction of these restricted solutions has only lower order effects on the overall solution dynamic). The result holds under a uniform bound on the advection coefficient, which we show to be sharp. We also extend it to models with non-local diffusion and KPP reactions.

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arXiv2022

Homogenization for Time-periodic KPP Reactions

We prove homogenization for reaction-advection-diffusion equations with KPP reactions, in the time-periodic spatially stationary ergodic setting, and find an explicit formula for the homogenized dynamic. We also extend this result to models with non-local diffusion and KPP reactions.

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arXiv2022

Homogenization for Space-Time-Dependent KPP Reaction-Diffusion Equations and G-Equations

We prove stochastic homogenization for reaction-advection-diffusion equations with random space-time-dependent KPP reactions with temporal correlations that are decaying in an appropriate sense. We show that the limiting homogenized dynamic has the simple form of spreading with some deterministic direction-dependent speeds from the support of the initial datum. We obtain analogous results for G-equations with random flame speeds and incompressible background advections. Important ingredients in our proofs are a non-autonomous subadditive theorem and the principle of virtual linearity for KPP reactions from the companion papers [30, 35].

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arXiv2021

Numerical Evidence of Exponential Mixing by Alternating Shear Flows

We performed a numerical study of the efficiency of mixing by alternating horizontal and vertical shear ``wedge'' flows on the two-dimensional torus. Our results suggest that except in cases where each individual flow is applied for only a short time, these flows produce exponentially fast mixing. The observed mixing rates are higher when the individual flow times are shorter (but not too short), and randomizing either the flow times or phase shifts of the flows does not appear to enhance mixing (again when the flow times are not too short). In fact, the latter surprisingly seems to inhibit it slightly.

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arXiv2021

Quantitative Homogenization for Combustion in Random Media

We obtain the first quantitative stochastic homogenization result for reaction-diffusion equations, for ignition reactions in dimensions $d\le 3$ that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton-Jacobi equations.

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arXiv2020

Long Time Dynamics for Combustion in Random Media

We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large space-time scale dynamics of solutions to these equations is almost surely governed by a different effective PDE, which should be a homogeneous Hamilton-Jacobi equation. While this was previously proved in one dimension as well as for isotropic reactions in several dimensions (i.e., with radially symmetric laws), we provide here the first proof of this phenomenon in the general non-isotropic multidimensional setting. Our results hold for reactions that have finite ranges of dependence (i.e., their values are independent at sufficiently distant points in space) as well as for some with infinite ranges of dependence, and are based on proving existence of deterministic front (propagation) speeds in all directions for these reactions.

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arXiv2020

Euler Equations on General Planar Domains

We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than $π$ and, in particular, is satisfied by all convex domains. The main ingredient in our approach is showing that constancy of the vorticity near the boundary is preserved for all time because Euler particle trajectories on these domains, even for general bounded solutions, cannot reach the boundary in finite time. We then use this to show that no vorticity can be created by the boundary of such possibly singular domains for general bounded solutions. We also show that our condition is essentially sharp in this sense by constructing domains that come arbitrarily close to satisfying it, and on which particle trajectories can reach the boundary in finite time. In addition, when the condition is satisfied, we find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary.

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arXiv2019

The Euler Equations in Planar Domains with Corners

When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the two-dimensional Euler equations has been established only in some special cases, and the solutions to which these results apply share the property that the diffuse part of the vorticity is constant near the points where the velocity is insufficiently regular. Assuming that the latter holds initially, the challenge is then to propagate this property along the Euler dynamic via an appropriate control of the Lagrangian trajectories. In domains with obtuse corners and sufficiently smooth elsewhere, Yudovich solutions fail to be almost Lipschitz only near these corners, and we investigate necessary and sufficient conditions for the vorticity to remain constant there. We show that if the vorticity is initially constant near the whole boundary, then it remains such forever (and global weak solutions are unique), provided no corner has angle greater than $π$. We also show that this fails in general for domains that do have such corners.

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