These lecture notes are devoted to solutions of hyperbolic-parabolic systems with persistent oscillations. We consider two examples both from mechanics: (i) The system of viscoelasticity of Kelvin-Voigt type with strain energies involving double well potentials, as employed in phase transitions. (ii) The compressible Navier-Stokes equations for a barotropic gas. For each system we construct solutions with persistent oscillations. In a later part we consider the nonlinear homogenization problem. For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids we show how ideas from the kinetic formulation of conservation laws can be used to derive effective equations. The effective equation consists by a kinetic equation coupled with the macroscopic flow.

We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. This estimate is crucial in the integrability analysis mean-field interactions induced by a singular kernel. The tensorial operator arises from a reformulation of integrability. Additionally, for smooth periodic solutions of the reformulated system, we derive a stability result.

In this paper we give a short and self-contained proof of the fact that weak solutions to the Maxwell-Stefan system automatically satisfy an entropy equality, establishing the absence of anomalous dissipation.

We consider the system of viscoelasticity with higher-order gradients and nonconvex energy in several space dimensions. We establish the asymptotic limits when the viscosity ν → 0 or when the dispersion coefficient δ → 0 . For the latter problem, it is worth noting that, for the case of two space dimensions, we also establish a rate of convergence. This result bears analogies to a result of Chemin (1996 Commun. PDE 21 1771–79) on the rate of convergence of the zero-viscosity limit for the two-dimensional Navier–Stokes equations with bounded vorticity.

We study Fokker–Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric is shown to emerge from an optimal control problem in the space of probability densities for a class of variable mobility matrices, with the cost function capturing the work dissipated via friction. Using the Nash-Kuiper isometric embedding theorem for Riemannian manifolds, we demonstrate the existence of optimal transport maps. Additionally, we construct a time-discrete variational scheme, establish key properties for the associated minimizing problem, and prove convergence to weak solutions of the associated Fokker-Planck equation.

We show that under natural growth conditions on the entropy function, convergence in relative entropy is equivalent to L_p-convergence. The main tool is the theory of Young measures, in a form that accounts for the formation of concentrations in weak limits.

A type-I model of non-isothermal multicomponent systems of gases describing mass diffusive and heat conductive phenomena is presented. The derivation of the model and a convergence result among thermomechanical theories in the smooth regime are discussed. Furthermore, the global-in-time existence of weak solutions and the weak-strong uniqueness property are established for the corresponding system with zero barycentric velocity.

In the first part of this paper we present some exact solutions for special hyperbolic-parabolic systems with sustained oscillations induced by the initial data, most notably the compressible Navier-Stokes system with non-monotone pressure. This part complements [A. E. Tzavaras, Sustained oscillations in hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 248 (2024) 51] where such examples are extensively studied. The second part deals with the problem of homogenization for one-dimensional models describing phase transitions for viscoelastic materials. Ideas from the kinetic formulation of conservation laws are employed to derive effective equations that describe the propagation of oscillations.

A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive definiteness of the matrix on a subspace and using the Bott--Duffin matrix inverse. The global existence of weak solutions and a weak-strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yielding $H^2(\Omega)$ bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

We study a one-dimensional nonlinear Vlasov equation with a local self-consistent force field generated by the density, where the force is given by the spatial derivative of a real-analytic nonlinearity. For small analytic initial data, we prove local-in-time existence and uniqueness of analytic solutions. In particular, this yields a perturbative well-posedness result around the trivial equilibrium. We also give an energy-based representation of weak stationary states and discuss perturbations around spatially homogeneous stationary profiles. The proof relies on a contraction mapping argument in a complete metric space of analytic functions. As a technical byproduct, we establish quantitative composition estimates for analytic nonlinearities in the analytic norms used in the argument.

We consider a set of bipolar Euler–Poisson equations and study two asymptotic limiting processes. The first is the zero-electron-mass limit, which formally results in a nonlinear adiabatic electron system. In a second step, we analyze the combined zero-electron-mass and quasi-neutral limits, which together lead to the compressible Euler equations. Using the relative energy method, we rigorously justify these limiting processes for weak solutions of the two-species Euler–Poisson equations that dissipate energy, as well as for strong solutions of the limit systems that are bounded away from vacuum. This justification is valid in the regime of initial data for which strong solutions exist. To deal with the electric potential, in the first case we use elliptic theory, whereas in the second case we employ the theory of Riesz potentials and properties of the Neumann function.

The Serre-Green-Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre-Green-Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that, over sufficiently extended time intervals, effectively approximates the classical Serre-Green-Naghdi equations and admits dispersive-diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre-Green-Naghdi equations and the effects of diffusion in the generation and propagation of undular bores.

Renewable energy forecasting services comprise various modules for intra-day and day-ahead forecasts. This work specifically addresses day-ahead forecasts, utilizing specifications based on endogenous, historical measurements. These specifications are designed to be computationally efficient, requiring fewer input variables and less training data. Such weather-independent specifications serve as benchmarks against the more computationally demanding forecasts based on numerical weather predictions. A series of experiments, designed to simulate the real-world application of an online system, were conducted on sliding windows of back-contact photovoltaic (installed at KAUST, Saudi Arabia) output series, solar irradiance recorded in Hawaii, and simulated data. Our analysis evaluated 24 specifications, which are variants of (i) functional time series models (including two novel shrinkage procedures); (ii) time series nearest neighbor schemes; (iii) exponential smoothing procedures; (iv) autoregressive integrated moving average processes; (v) automatic techniques based on time series decomposition; and (vi) the persistence model. In addition to employing outlier-robust accuracy metrics, such as mean absolute error, our evaluation also prioritized prediction-interval accuracy, quantified by the mean scaled interval score. Our findings suggest that practitioners can achieve significant improvements over the persistence model by forecasting daily profiles using adaptive nonparametric or functional data analysis-based procedures. Moreover, applying shrinkage to nearest neighbor (NN) forecasts toward smooth, average daily profiles significantly enhances NN performance. Conversely, some popular, computationally intensive models fail to perform adequately to justify their additional cost.

The derivation of an approximate Class-I model for nonisothermal multicomponent systems of fluids, as the high-friction limit of a Class-II model is justified, by validating the Chapman-Enskog expansion performed from the Class-II model towards the Class-I model. The analysis proceeds by comparing two thermomechanical theories via relative entropy.

We consider an infinite vortex line in a fluid which interacts with a boundary surface as a simplified model for tornadoes. We study self-similar solutions for stationary axisymmetric Navier-Stokes equations and investigate the types of motion which are compatible with this structure when viscosity is non-negative. For viscosity equal to zero, we construct a class of explicit stationary solutions. We then consider solutions with slip discontinuity and show that they do not exist in this framework.

This note introduces the relative energy for two-species electromagnetic fluid models. The relative energy identity for a bipolar Euler-Maxwell system is formally derived. Moreover, two applications of the relative energy for these type of systems are given. The first application deals with the weak-strong uniqueness property of a Euler-Poisson system, while the second concerns the relaxation limit of a bipolar Euler-Poisson system towards a bipolar drift-diffusion system.

We consider the incompressible axisymmetric Navier-Stokes equations with swirl as an idealized model for tornado-like flows. Assuming an infinite vortex line which interacts with a boundary surface resembles the tornado core, we look for stationary self-similar solutions of the axisymmetric Euler and axisymmetric Navier-Stokes equations. We are particularly interested in the connection of the two problems in the zero-viscosity limit. First, we construct a class of explicit stationary self-similar solutions for the axisymmetric Euler equations. Second, we consider the possibility of discontinuous solutions and prove that there do not exist self-similar stationary Euler solutions with slip discontinuity. This nonexistence result is extended to a class of flows where there is mass input or mass loss through the vortex core. Third, we consider solutions of the Euler equations as zero-viscosity limits of solutions to Navier-Stokes. Using techniques from the theory of Riemann problems for conservation laws, we prove that, under certain assumptions, stationary self-similar solutions of the axisymmetric Navier-Stokes equations converge to stationary self-similar solutions of the axisymmetric Euler equations as $\nu\to0$. This allows to characterize the type of Euler solutions that arise via viscosity limits.

We construct examples of oscillating solutions with persistent oscillations for various hyperbolic-parabolic systems with singular diffusion matrices that appear in mechanics. These include an example for the equations of nonlinear viscoelasticity of Kelvin–Voigt type with stored energy that violates rank-one convexity, which amounts to a time-dependent variant of twinning solutions. We also present an example pertaining to the system of gas dynamics with thermal effects for a viscous, adiabatic gas. Finally, we show an example for the compressible Navier–Stokes system in one-space dimension with nonmonotone pressure function. We also study the existence of oscillating solutions for linear hyperbolic-parabolic systems with singular diffusion matrices.

A Type-I model of a multicomponent system of fluids with non-constant temperature is derived as the high-friction limit of a Type-II model via a Chapman-Enskog expansion. The asymptotic model is shown to fit into the general theory of hyperbolic-parabolic systems, by exploiting the entropy structure inherited through the asymptotic procedure. Finally, by deriving the relative entropy identity for the Type-I model, two convergence results for smooth solutions are presented, from the system with mass-diffusion and heat conduction to the corresponding system without mass-diffusion but including heat conduction and to its hyperbolic counterpart.

We give conditions that guarantee uniqueness of renormalized solutions for the Maxwell-Stefan system. The proof is based on an identity for the evolution of the symmetrized relative entropy. Using the method of doubling the variables we derive the identity for two renormalized solutions and use information on the spectrum of the Maxwell-Stefan matrix to estimate the symmetrized relative entropy and show uniqueness.