I defended my PhD in mathematics at University of Warsaw (Poland) in 2015. My research experience includes internships at University of Heidelberg (Germany) and postdoc positions at University of Warwick (UK), University of Basel (Switzerland) and Institute of Mathematics, Polish Academy of Sciences in Warsaw. I am passionate about natural sciences, traveling, learning natural languages, cycling and hiking in the mountains (especially in the Alps).
My scientific interests revolve around mathematical modeling and analysis of real-world systems. I have developed and studied models of the biological process of cell differentiation and studied partial differential models originating in fluid mechanics. In 2023 I joined COeXISTENCE at Jagiellonian University where I am leveraging my skills and expertise in the field of artificial intelligence in transportation. My current interests are focused on human and machine learning as well as human-machine interaction.
List of main publications and preprints
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On measures of accretion and dissipation for solutions of the Camassa-Holm equation
Jamróz, Grzegorz
Journal of Hyperbolic Differential Equations
2017
We investigate the measures of dissipation and accretion related to the weak solutions of the Camassa–Holm equation. Demonstrating certain novel properties of nonunique characteristics, we prove a new representation formula for these measures and conclude about their structural features, such as the fact that they are singular with respect to the Lebesgue measure. We apply these results to gain new insights into the structure of weak solutions, proving in particular that measures of accretion vanish for dissipative solutions of the Camassa–Holm equation.
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Transfer of energy in Camassa-Holm and related models by use of nonunique characteristics
Jamróz, Grzegorz
Journal of Differential Equations
2017
We study the propagation of energy density in finite-energy weak solutions of the Camassa–Holm and related equations. Developing the methods based on generalized nonunique characteristics, we show that the parts of energy related to positive and negative slopes are one-sided weakly continuous and of bounded variation, which allows us to define certain measures of dissipation of both parts of energy. The result is a step towards the open problem of uniqueness of dissipative solutions of the Camassa–Holm equation.
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On uniqueness of dissipative solutions of the Camassa-Holm equation
Jamróz, Grzegorz
arXiv preprint arXiv:1611.00333
2016
We prove that dissipative weak solutions of the Camassa-Holm equation are unique. Thus we complete the global well-posedness theory of this celebrated model of shallow water, initiated by a general proof of existence in [Z. Xin, P. Zhang Comm. Pure Appl. Math. 53 (2000)]. As the dissipative weak solutions, being viscosity solutions, seem to constitute the most physically relevant class of solutions in the wave-breaking regime, the result provides mathematical rationale for the feasibility of the Camassa-Holm equation as a model of water waves encompassing both soliton interactions and wave-breaking.
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Measure-transmission metric and stability of structured population models
Jamróz, Grzegorz
Nonlinear Analysis: Real World Applications
2015
In Gwiazda, et al. (2012) a framework for studying cell differentiation processes based on measure-valued solutions of transport equations was introduced. Under application of the so-called measure-transmission conditions it enabled to describe processes involving both discrete and continuous transitions. This framework, however, admits solutions which lack continuity with respect to initial data. In this paper, we modify the framework from Gwiazda, et al. (2012) by replacing the flat metric, known also as bounded Lipschitz distance, by a new Wasserstein-type metric. We prove, that the new metric provides stability of solutions with respect to perturbations of initial data while preserving their continuity in time. The stability result is important for numerical applications.
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Nonnegative measures belonging to H- 1 (R2)
Jamróz, Grzegorz
Comptes Rendus Mathematique
2015
Motivated by applications in fluid dynamics, we show elementarily that a nonnegative compactly supported Radon measure μ belongs to the negative Sobolev space 𝐻−1(𝑅2) provided that function 𝑟↦𝜇(𝐵(0,𝑟)) is Hölder continuous. In passing we obtain embedding of the space of nondecreasing Hölder continuous functions on 𝑅 into the fractional Sobolev space 𝐻1/2(𝑅). We comment on possible generalizations and numerical applications.
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Elliptic equations, multivalued x-dependent graphs and Orlicz spaces
Jamróz, Grzegorz
Nonlinear Analysis: Theory, Methods & Applications
2010
