The Ukraine-NASTRAN Winter School “Nonlinear Partial Differential Equations” offers six lecture courses covering a range of methods and theories of nonlinear partial differential equations (PDEs). Topics include integrable systems, continuation beyond wave breaking, elliptic and degenerate parabolic equations, fractional differential equations, stochastic fluid flows, and compensated compactness for conservation laws.
Each course consists of three lectures. The courses are delivered by researchers affiliated with international universities and research institutions.
Compensated Compactness for Scalar Conservation Laws
Giuseppe Maria Coclite
Politecnico di Bari
Course abstract
Compensated compactness is a useful measure-theoretic tool based on the compression effects of nonlinear equations. It works only in the nonlinear case and applies to several convergence problems.
Lecture topics
Motivation
Compensated compactness
Young measures
Murat's Lemma
Div–Curl Lemma
Proof of Compensated Compactness
Global Existence of Bounded Solutions
Long Time Behavior of Periodic Solutions
Diffusive-Dispersive Limits
Stochastic Fluid Flows
Konstantina Trivisa
University of Maryland
Lecture 1. Invariant Measures for the Stochastic Navier–Stokes Equations for Compressible Flows and the Problem of Turbulence
Abstract
The lecture presents results on the long-time behaviour of solutions to a stochastically forced Navier–Stokes system describing the motion of a compressible viscous fluid. In the one-dimensional case, the existence of an invariant measure for the Markov process generated by strong solutions was established in collaboration with Coti-Zelati and Glatt-Holtz.
The work addresses the difficulties associated with non-Feller Markov semigroups on non-complete metric spaces by generalizing the classical Krylov–Bogoliubov method and providing suitable polynomial and exponential moment bounds on the solution, together with pathwise estimates. The lecture concludes with a discussion of recent developments in the multidimensional case for related models.
Reference
M. Coti Zelati, N. Glatt-Holtz, and K. Trivisa, “Invariant Measures for the Stochastic One-Dimensional Compressible Navier-Stokes Equations.” Journal of Applied Mathematics and Optimization, 83 (2021), no. 3, 1487–1522.
Lecture 2. Statistically Stationary Solutions to the Stochastic Isentropic Compressible Euler Equations with Linear Damping
Abstract
This lecture investigates the long-time behaviour of isentropic compressible Euler equations with linear damping driven by white-in-time noise on a one-dimensional torus.
The authors establish the existence of a statistically stationary solution, meaning a solution whose law is invariant under the stochastic dynamics of the PDE, in the class of weak martingale entropy solutions for any adiabatic constant γ > 1 and any damping coefficient α > 0. The solution satisfies an associated entropy inequality.
The result is established using a multi-level approximation scheme involving a truncation parameter R and an artificial viscosity parameter ε.
Reference
J. Kuan, K. Tawri, and K. Trivisa, “Statistically stationary solutions to the stochastic isentropic compressible Euler equations with linear damping.” Under revision, JMPA (2026).
Lecture 3. Anomalous Dissipation for Stochastic Fluid Models
Tentative topic
Abstract
A prevalent feature of three-dimensional turbulence is anomalous dissipation, in which the mean rate of energy dissipation is bounded below by a positive number in the inviscid limit. This is thought to be due to the nonlinear convection term in the Navier–Stokes equations stretching vortex tubules and thereby increasing the amount of small-scale oscillations within the flow.
The work constructs an example of a linear Stokes flow within a sphere that exhibits anomalous dissipation.
Reference
Ethan Dudley and Konstantina Trivisa, “The Existence of Anomalous Dissipation over Bounded Interior Domains.” Preprint (2025), arXiv:2501.00705.
Riemann-Hilbert Problems and Applications to Integrable PDE
Dmitry Shepelsky
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
Course abstract
In contrast with linear PDEs, there is no general theory of nonlinear partial differential equations. However, for certain classes of nonlinear PDEs, it is possible to construct many exact solutions and efficiently solve general problems of specific types, such as Cauchy (initial-value) problems.
These equations are known as integrable equations or soliton equations because they support localized, stable nonlinear travelling waves called solitons, which retain their shape and speed after interactions.
The lectures introduce the basics of Riemann–Hilbert problems, which are boundary-value problems for piecewise analytic functions in the complex plane. They discuss the Lax pair representation, a fundamental property characterizing integrable PDEs, and explain how solutions to Cauchy problems can be represented in terms of appropriate Riemann–Hilbert problems.
The nonlinear Schrödinger equation serves as a prototype model of nonlinear wave propagation. The course also considers the nonlinear steepest descent method, which provides an effective way to study the long-time behaviour of solutions to Cauchy problems for integrable equations.
Stampacchia's Lemma and Application in PDE Analysis
Roman Taranets
Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine; Yuriy Fedkovych Chernivtsi National University
Course abstract
This advanced course focuses on truncation methods and Stampacchia's classical lemmas, which serve as powerful tools in the qualitative analysis of linear and nonlinear partial differential equations.
Participants will learn techniques for deriving (L^\infty)-bounds for elliptic equations with measurable coefficients and apply these methods to investigate qualitative properties of degenerate parabolic systems, using the porous medium equation as a primary model.
Lecture syllabus
1. Stampacchia's Lemmas
Formulation, proof, and geometric interpretation of Stampacchia's foundational lemmas on differential inequalities. Techniques involving energy estimates and De Giorgi–Stampacchia-type integral inequalities.
2. Applications to Elliptic Equations: Boundedness of Solutions
Application of Stampacchia's lemmas to prove the boundedness of solutions to divergence-form elliptic equations with rough (measurable) coefficients. Transitioning from energy estimates to (L^\infty)-regularity.
3. Application to the Porous Medium Equation: Finite Speed of Propagation and Waiting Time Phenomenon
Analysis of degenerate parabolic equations. Investigating local solution behaviour, proving the finite speed of propagation property for disturbances, and analysing the waiting-time phenomenon.
Introduction to Fractional Differential Equations
Nataliya Vasylyeva
S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine; Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine
Course abstract
The lectures cover selected topics in fractional calculus, the theory of differential and integral operators of non-integer order and, in particular, differential equations containing such operators.
Course topics
Definitions of fractional integrals and fractional derivatives and their properties.
Fourier and Laplace transformations of nonlocal operators.
Micro- and macrolevel approaches to deriving fractional differential equations from physical laws.
Examples of models in mechanics, including the theory of viscoelasticity, and thermal disease diagnostics, including bioheat transfer in living tissues described by fractional PDEs.
Single-term Caputo fractional differential equations: existence, uniqueness, smoothness of the solution, and the influence of perturbation data.
Solutions of Nonlinear PDE Beyond Blow-Up
Yan Rybalko
University of Oslo; B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
Course abstract
The course discusses the continuation of solutions of a certain class of nonlinear, hyperbolic-like partial differential equations beyond finite-time blow-up. The Camassa–Holm (CH) equation is used as a prototype.
The CH equation is one of the principal models for describing the propagation of shallow-water waves and incorporates nonlinear phenomena such as soliton interaction and wave breaking. Wave breaking occurs when the slope of the solution becomes infinite in finite time while the solution itself remains uniformly bounded.
The lectures discuss the exact non-regular solitons known as peakons, which are weak solutions of the equation, and show how peakon interactions can cause wave breaking. By applying the method of characteristics and using the Bressan–Constantin variables, the course establishes a global solution of the general Cauchy problem for the Camassa–Holm equation that survives possible finite-time wave breaking.