Measure-valued Solutions Research Articles

Cucker-Smale model with normalized communication weights and time delay

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity. For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.

Asymptotic Behavior of a Critical Fluid Model for a Processor Sharing Queue via Relative Entropy

In this paper, we develop a new approach to studying the asymptotic behavior of fluid model solutions for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. In contrast to the approach used in [12], which does not readily generalize to networks of processor sharing queues, we expect the approach developed in this paper to be more robust. Indeed, we anticipate that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under various resource sharing protocols naturally described by measure-valued processes.

Semigeostrophic equations in physical space with free upper boundary

We define Lagrangian solutions in physical space for 3-d incompressible semigeostrophic system with free upper boundary under various conditions for initial data, then prove their existence via the minimization with respect to a geostrophic functional, generalizing the results of Cullen and Feldman (J Math Anal 37(5): 1371–1395, 2006) and Feldman and Tudorascu (Arch Ration Mech Anal 218(1): 527–551 2015) to the situation of free upper boundary. As a byproduct of our proof, we obtain the existence of measure-valued dual space solutions when the initial measure $$\nu _0\in \mathcal {P}_2(\mathbb {R}^3)$$ and is supported on $$\{-\frac{1}{\delta }\le y_3\le -\delta \}$$ .

Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

Dissipative measure-valued solutions to the compressible Navier–Stokes system

We introduce a new concept of dissipative measure-valued solution to the compressible Navier–Stokes system satisfying, in addition, a relevant form of the total energy balance. Then we show that a dissipative measure-valued and a standard smooth classical solution originating from the same initial data coincide (weak-strong uniqueness principle) as long as the latter exists. Such a result facilitates considerably the proof of convergence of solutions to various approximations including certain numerical schemes that are known to generate a measure-valued solution. As a byproduct we show that any measure-valued solution with bounded density component that starts from smooth initial data is necessarily a classical one.

On the computation of measure-valued solutions

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm ofmeasure-valued solutionsfor these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

Singular Perturbations of Forward-Backward p-Parabolic Equations

In this paper we have proved the existence of entropy measure-valued solutions to forward-backward p-parabolic equations. We have obtained these solutions as singular limits of weak solutions to (p,q)-elliptic regularized boundary-value problems as e → 0+. When q > 1 and q = 2 we have not defined yet admissible initial and final conditions even in the form of integral inequalities.

On a stochastic nonlocal conservation law in a bounded domain

In this paper, we are interested in the Dirichlet boundary value problem for a multi-dimensional nonlocal conservation law with a multiplicative stochastic perturbation in a bounded domain. Using the concept of measure-valued solutions and Kruzhkov's semi-entropy formulations, a result of existence and uniqueness of entropy solution is proved.

Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws

Entropy stability plays an important role in the dynamics of nonlinear systems of hyperbolic conservation laws and related convection-diffusion equations. Here we are concerned with the corresponding question of numerical entropy stability --- we review a general framework for designing entropy stable approximations of such systems. The framework, developed in [28,29] and in an ongoing series of works [30,6,7], is based on comparing numerical viscosities to certain entropy-conservative schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks. &nbsp We demonstrate this approach with a host of second-- and higher--order accurate schemes, ranging from scalar examples to the systems of shallow-water, Euler and Navier-Stokes equations. We present a family of energy conservative schemes for the shallow-water equations with a well-balanced description of their steady-states. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in Euler equations, and we conclude with the computation of entropic measure-valued solutions based on the class of so-called TeCNO schemes --- arbitrarily high-order accurate, non-oscillatory and entropy stable schemes for systems of conservation laws.

Condensation phenomena in nonlinear drift equations

We study non-negative, measure-valued solutions to nonlinear drifttype equations modelling concentration phenomena related to Bose-Einstein particles. In one spatial dimension, we prove existence and uniqueness for measure solutions. Moreover, we prove that all solutions blow up in finite time leading to a concentration of mass only at the origin, and the concentrated mass absorbs increasingly the mass converging to the total mass as t → ∞. Our analysis makes a substantial use of independent variable scalings and pseudo-inverse functions techniques.

Nonuniqueness of solutions for a class of forward–backward parabolic equations

We study the initial–boundary value problem {ut=[φ(u)]xx+ε[ψ(u)]txxinΩ×(0,∞)φ(u)+ε[ψ(u)]t=0in∂Ω×(0,∞)u=u0≥0inΩ×{0} with measure-valued initial data. Here Ω is a bounded open interval, φ(0)=φ(∞)=0, φ is increasing in (0,α) and decreasing in (α,∞), and the regularising term ψ is increasing but bounded. It is natural to study measure-valued solutions since singularities may appear spontaneously in finite time. Nonnegative Radon measure-valued solutions are known to exist and their construction is based on an approximation procedure. Until now nothing was known about their uniqueness.In this note we construct some nontrivial examples of solutions which do not satisfy all properties of the constructed solutions, whence uniqueness fails. In addition, we classify the steady state solutions.

On the Convergence of a Shock Capturing Discontinuous Galerkin Method for Nonlinear Hyperbolic Systems of Conservation Laws

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.

Probability Measures with Finite Moments and the Homogeneous Boltzmann Equation

We characterize the class of probability measures possessing finite moments of an arbitrary positive order in terms of the symmetric difference operators of their Fourier transforms. As an application, we prove the continuity of probability densities associated with measure-valued solutions to the Cauchy problem for the homogeneous Boltzmann equation with Maxwellian molecules.

The Escalator Boxcar Train method for a system of age-structured equations

The Escalator Boxcar Train method (EBT) is a numerical method for structured population models of McKendrick -- von Foerster type. Those models consist of a certain class of hyperbolic partial differential equations and describe time evolution of the distribution density of the structure variable describing a feature of individuals in the population. The method was introduced in late eighties and widely used in theoretical biology, but its convergence was proven only in recent years using the framework of measure-valued solutions. Till now the EBT method was developed only for scalar equation models. In this paper we derive a full numerical EBT scheme for age-structured, two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. It is the first step towards extending the EBT method to systems of structured population equations.

Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes \emph{may not} necessarily converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. Furthermore, we propose a more general notion, that of \emph{entropy measure valued solutions}, as an appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.

Long Time Convergence of the Bose–Einstein Condensation

We study long time behavior of the Bose–Einstein condensation of measure-valued solutions \(F_t\) of the space homogeneous and velocity isotropic Boltzmann equation for Bose–Einstein particles at low temperature. We prove that if \(F_0\ge 0\) is a non-singular Borel measure on \({{\mathbb R}}_{\ge 0}\) satisfying a very low temperature condition and that the ratio \(F_0([0,\varepsilon ])/\varepsilon ^{\alpha }\) is sufficiently large for all \(\varepsilon \in (0, R]\) for some constants \(0 0\), then there exists a solution \(F_t\) of the equation on \([0,+\infty )\) with the initial datum \(F_0\) such that \(F_t(\{0\})\) converges to the expected Bose–Einstein condensation as \(t\rightarrow +\infty \). We also show that such initial data \(F_0\) exist extensively.

Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage–Hutter model of granular flows in one and two space dimensions. For the latter system, we also show the complete dissipation of momentum in finite time, thus rigorously justifying an assumption that has been made in the engineering and numerical literature.

Measure valued solutions to the stochastic Euler equations in $$\mathbb {R}^d$$ R d

We establish the existence of measure-valued solutions of the stochastic Euler equations in \(\mathbb {R}^d, \, d \ge 2.\) Our definition of a measure-valued solution is based on that for the deterministic Euler equations. Our first result is the existence over a given probability space in the spirit of Bensoussan and Temam (J Funct Anal 13:195–222, 1973). The main tool is measurable selection established in Stroock and Varadhan (Multidimensional diffusion processes, 1997). As a necessary step for this result, we also present a new existence result on the stochastic Navier–Stokes equations. Our second result is the existence of a martingale measure-valued solution. The main tool is the generalized Skorokhod representation theorem due to Jakubowski (Theory Probab Appl, 42:167–174, 1998).

Conservation laws driven by Lévy white noise

We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itó–Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first prove the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the L1-contraction principle. Finally, the L1 contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.

Computation of measure-valued solutions for the incompressible Euler equations

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure-valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure-valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure-valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two-dimensional vortex sheet, which indicates that the computed measure-valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort.