# Euler Alignment, Nonlinear Conservation Laws and the Pressure-less System

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of Maryland, College Park (MD) · $300,000

## Abstract

Partial Differential Equations play a pivotal role in a wide range of applications, facilitating the study of many questions in physics, geometry, meteorology, biology, economics, and engineering sciences, to name a few. This project aims to advance the current understanding of fundamental questions that arise in the context of three canonical classes of nonlinear partial differential equations, which model (i) emergent phenomena; (ii) conservation laws; and (iii) the pressure-less early universe model.  While these three classes of evolution equations are well-understood in the one-dimensional spatial setting, the questions of existence, regularity and large-time behavior of solutions for the more realistic multi-dimensional models are mostly open. The plan of this project is to develop novel paradigms to address these questions with emphasis on the multi-dimensional setting. This project also involves mentoring graduate students who will be involved in this research.
	
This project is concerned with the following time-dependent partial differential equations in multiple spatial dimensions. (i) Euler Alignment. The system of Euler Alignment arises as the large crowd dynamics of the Cucker-Smale alignment model. The goal is to study the open question of existence of multidimensional strong solutions, subject to sub-critical initial data, and their large-time behavior with short-range communication kernels. (ii) Nonlinear Conservation Laws. Nonlinear scalar conservation laws

## Key facts

- **NSF award ID:** 2508407
- **Awardee organization:** University of Maryland, College Park (MD)
- **SAM.gov UEI:** NPU8ULVAAS23
- **PI:** Eitan Tadmor
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $300,000
- **Funds obligated:** $300,000
- **Transaction type:** Standard Grant
- **Period:** 07/01/2025 → 06/30/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2508407

## Citation

> US National Science Foundation, Award 2508407, Euler Alignment, Nonlinear Conservation Laws and the Pressure-less System. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2508407. Licensed CC0.

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