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Grzegorz Karch

Publications and source records attributed to Grzegorz Karch.

At least 19 recordsLinked to original sources
arXiv2026

Aggregation--Diffusion Equations with Regular or Repulsive Interactions: Gaussian and Self-Similar Asymptotics

We study the large-time behavior of solutions to the aggregation--diffusion equation $$ u_t=Δu+\nabla\cdot\bigl(u\,\nabla K*u\bigr) \qquad \text{in } \mathbb{R}^d. $$ Our main results identify a transition, governed by the regularity and singularity of the interaction kernel, between Gaussian and nonlinear self-similar asymptotics. We prove that for regular interaction kernels, as well as for sufficiently mild singular repulsive kernels, the nonlinear drift is asymptotically negligible and solutions exhibit the same Gaussian large-time behavior as solutions of the linear heat equation. In contrast, for the critical logarithmic kernel $ K(x)=-\log |x|, $ the interaction persists at the diffusive scale and leads to genuinely nonlinear asymptotics. More precisely, for every mass \(M>0\), we prove the existence and uniqueness of a self-similar solution of mass \(M\) and show that every solution with mass $M$ and with finite second moment converges in all $L^p$-norms to this self-similar solution as \(t\to\infty\).

math.AP↗
arXiv2026

Nemotron-Labs-3-Puzzle-75B-A9B: Compressing Hybrid MoE LLMs

We present Nemotron-Labs-3-Puzzle-75B-A9B, a compressed variant of Nemotron-3-Super optimized for interactive deployment. We designed the model to maximize server throughput under high user throughput constraints. In interactive serving workloads on a single 8xB200 node, Puzzle-75B-A9B achieves approximately 2x higher server throughput than Nemotron-3-Super at matched user throughput constraints. In ultra-long-context deployment on a single H100 GPU, the compressed model increases 1M-token concurrency from 1 request to 8 requests. Puzzle-75B-A9B is constructed using a multi-stage pipeline that combines the Iterative Puzzle compression framework with knowledge distillation, reinforcement learning, quantization, and a Multi-Token Prediction head. The compression process jointly optimizes heterogeneous MoE pruning, active parameter budget, and Mamba pruning to improve inference efficiency while preserving model quality. We evaluate Puzzle-75B-A9B on a broad suite of reasoning, coding, multilingual, long-context, and agentic benchmarks. Despite substantial compression, the model retains strong downstream accuracy relative to the parent model across a wide range of tasks. These results demonstrate that large hybrid MoE models can be substantially optimized for deployment efficiency while maintaining strong downstream capability. Our model is publicly available on Hugging Face.

cs.AI↗
arXiv2026

Concentration of mass of solutions to aggregation-diffusion equations

We consider the aggregation-diffusion equation in the whole space with a mildly singular interaction kernel K = K(x) which behaves like |x|^k near the origin for some k $\in$ (0, 2). This equation, supplemented with nonnegative, bounded, and integrable initial data, possesses a global-in-time solution. We prove that the family of nonnegative, radially symmetric solutions of this equation, all sharing the same initial datum, focuses around the origin over a common finite time interval as $ε$ ___ 0.

math.AP↗
arXiv2025

Tools for stability analysis of fractional reaction diffusion systems

The linearization principle states that the stability (or instability) of solutions to a suitable linearization of a nonlinear problem implies the stability (or instability) of solutions to the original nonlinear problem. In this work, we prove this principle for solutions of abstract fractional reaction-diffusion equations with a fractional derivative in time of order $α\in (0,1)$. Then, we apply these results to particular fractional reaction-diffusion equations, obtaining, for example, the counterpart of the classical Turing instability in the case of fractional equations.

math.AP↗
arXiv2025

Large self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field

The Navier-Stokes system for an incompressible fluid coupled with the equation for a heat transfer is considered in the whole three dimensional space. This system is invariant under a suitable scaling. Using the Leray-Schauder theorem and compactness arguments, we construct self-similar solutions to this system without any smallness assumptions imposed on homogeneous initial conditions.

math.AP↗
arXiv2024

Solutions at vacuum and rarefaction waves in pressureless Euler alignment system

We construct global-in-time weak solutions to the pressureless Euler alignment system posed on the whole line and supplemented with initial conditions, where an initial density is an arbitrary, nonnegative, bounded, and integrable function (hence density at vacuum is allowed) and the corresponding initial velocity is determined by certain inequalities. Moreover, our setting covers the case where solutions to the pressureless Euler alignment system are known to be non-smooth. We also study an asymptotic behavior of constructed solutions and we show that, under a suitable rescaling, the density looks like a uniform distribution on a bounded, time dependent, expanding-in-time interval and the corresponding velocity approaches a rarefaction wave (i.e. the well-known explicit solution to the inviscid Burgers equation).

math.AP↗
arXiv2023

Stable discontinuous stationary solutions to reaction-diffusion-ODE systems

A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial boundary value problems may have different types of stationary solutions. In our parallel work [Instability of all regular stationary solutions to reaction-diffusion-ODE systems (2021)], regular (i.e. sufficiently smooth) stationary solutions are shown to exist, however, all of them are unstable. The goal of this work is to construct discontinuous stationary solutions to general reaction-diffusion-ODE systems and to find sufficient conditions for their stability.

math.AP↗
arXiv2022

Discontinuous stationary solutions to certain reaction-diffusion systems

Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the Brusselator model, the Gray-Scott model, the Oregonator model and a certain predator-prey model. It is shown that the considered systems have the both smooth and discontinuous stationary solutions, however, only discontinuous ones can be stable.

math.AP↗
arXiv2021

Instability of all regular stationary solutions to reaction-diffusion-ODE systems

A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value problems may have regular (i.e. sufficiently smooth) stationary solutions. This class of {\it close-to-equilibrium} patterns includes stationary solutions that emerge due to the Turing instability of a spatially constant stationary solution. The main result of this work is instability of all regular patterns. It suggests that stable stationary solutions arising in models with non-diffusive components must be {\it far-from-equilibrium} exhibiting singularities. Such discontinuous stationary solutions have been considered in our parallel work [\textit{Stable discontinuous stationary solutions to reaction-diffusion-ODE systems}, preprint (2021)].

math.AP↗
arXiv2021

Mathematical treatment of PDE model describing chemotactic E. coli colonies

We consider an initial-boundary value problem describing the formation of colony patterns of bacteria Escherichia coli. This model consists of reaction-diffusion equations coupled with the Keller-Segel system from the chemotaxis theory in a bounded domain, supplemented with zero-flux boundary conditions and with non-negative initial data. We answer questions on the global in time existence of solutions as well as on their large time behaviour. Moreover, we show that solutions of a related model may blow up in a finite time.

math.AP↗
arXiv2021

Stability of constant steady states of a chemotaxis model

The Cauchy problem for the parabolic--elliptic Keller--Segel system in the whole $n$-dimensional space is studied. For this model, every constant $A \in \mathbb{R}$ is a stationary solution. The main goal of this work is to show that $A < 1$ is a stable steady state while $A > 1$ is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.

math.AP↗
arXiv2020

Stability of singular solutions to the Navier-Stokes system

We develop mathematical methods which allow us to study asymptotic properties of solutions to the three dimensional Navier-Stokes system for incompressible fluid in the whole three dimensional space. We deal either with the Cauchy problem or with the stationary problem where solutions may be singular due to singular external forces which are either singular finite measures or more general tempered distributions with bounded Fourier transforms. We present results on asymptotic properties of such solutions either for large values of the space variables (so called the far-field asymptotics) or for large values of time.

math.AP↗
arXiv2020

Sharp Sobolev estimates for concentration of solutions to an aggregation-diffusion equation

We consider the drift-diffusion equation $u_t-εΔu + \nabla \cdot(u\nabla K^*u)=0$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case $K(x)=-|x|$. We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity $ε$ studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.

math.AP↗
arXiv2020

A framework for non-local, non-linear initial value problems

We study the Cauchy problem for non-linear non-local operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g. the porous medium equation with the fractional Laplacian and the parabolic fractional $p$-Laplacian. We show the existence, uniqueness of bounded solutions and study their further properties. Several new examples of non-local, non-linear operators are provided.

math.AP↗
arXiv2020

Concentration phenomena in a diffusive aggregation model

We consider the drift-diffusion equation $$ u_t-\varepsilon Δu+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation with small diffusivity are studied.

math.AP↗
arXiv2017

Dynamical spike solutions in a nonlocal model of pattern formation

Coupling a reaction-diffusion equation with ordinary differential equations (ODE) may lead to diffusion-driven instability (DDI) which, in contrast to the classical reaction-diffusion models, causes destabilization of both, constant solutions and Turing patterns. Using a shadow-type limit of a reaction-diffusion-ODE model, we show that in such cases the instability driven by nonlocal terms (a counterpart of DDI) may lead to formation of unbounded spike patterns.

math.AP↗