Papers by Cleopatra Christoforou
Mathematical Models and Methods in Applied Sciences, Oct 1, 2022
We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space... more We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions with concentration to the Cauchy problem for any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions with concentration admit time-asymptotic flocking.
arXiv (Cornell University), Mar 1, 2023
We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugat... more We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into R 3. The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a C 1,1 isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in L 2 .
arXiv (Cornell University), Aug 25, 2018
For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measur... more For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. However, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measurevalued weak versus strong uniqueness using the averaged relative entropy inequality.
Corrugated versus smooth uniqueness and stability of negatively curved isometric immersions
Quarterly of Applied Mathematics
We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugat... more We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into R 3 \mathbb {R}^3 . The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a C 1 , 1 C^{1,1} isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in L 2 L^2 .
Elastic medium: the flux F is determined by the value U(x, t). Viscoelastic medium: the flux depe... more Elastic medium: the flux F is determined by the value U(x, t). Viscoelastic medium: the flux depends also on the past history of the medium U(x, τ) for τ < t. Materials with fading memory that correspond to constitutive relations with flux of the form: F (U(x, t)) + ∫ t 0 k(t − τ)G(U(x, τ)) dτ (1) i.e. Ut+ F (U)x+ ∫ t
Decay of Positive Waves of Hyperbolic Balance Laws
Acta Mathematica Scientia, 2012
ABSTRACT Historically, decay rates have been used to provide quantitative and qualitative informa... more ABSTRACT Historically, decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time. This work presents two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics [7, 17, 24].
We consider a 2×2 system of hyperbolic balance laws, in one-space dimension, that describes the e... more We consider a 2×2 system of hyperbolic balance laws, in one-space dimension, that describes the evolution of a granular material with slow erosion and deposition. The dynamics is expressed in terms of the thickness of a moving layer on top and of a standing layer at the bottom. The system is linearly degenerate along two straight lines in the phase plane and genuinely nonlinear in the subdomains confined by such lines. In particular, the characteristic speed of the first characteristic family is strictly increasing in the region above the line of linear degeneracy and strictly decreasing in the region below such a line. The non dissipative source term is the product of two quantities that are transported with the two different characteristic speeds. The global existence of entropy weak solutions of the Cauchy problem for such a system was established by Amadori and Shen [3] for initial data with bounded but possibly large total variation, under the assumption that the initial height of the moving layer be sufficiently small. In this paper we establish the Lipschitz L 1-continuous dependence of the solutions on the initial data with a Lipschitz constant that grows exponentially in time. The proof of the L 1-stability of solutions is based on the construction of a Lyapunov like functional equivalent to the L 1-distance, in the same spirit of the functional introduced by Liu and Yang [40] and then developed by Bressan, Liu, Yang [19] for systems of conservation laws with genuinely nonlinear or linearly degenerate characteristic fields.
Springer Proceedings in Mathematics & Statistics, 2018
The work of [5] on the extension of the relative entropy identity to the class of hyperbolic-para... more The work of [5] on the extension of the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable is the context of this article. The general theory is presented and the derivation of the relative entropy identities for both hyperbolic and hyperbolic-parabolic systems is presented. The resulting identities are useful to provide measure valued weak versus strong uniqueness theorems as well as convergence results in the zero-viscosity limit. An application of this theory is given for the example of the system of thermoviscoelasticity.
We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperb... more We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an L^p framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.
We deal with the viscous approximation of a system of conservation laws in one space dimension an... more We deal with the viscous approximation of a system of conservation laws in one space dimension and we focus on initial-boundary value problems. It is known that, in general, different viscous approximation provide different limits because of boundary layer phenomena. We focus on Riemann-type data and we discuss a uniqueness criterion for distributional solutions which applies to both the non characteristic and the boundary characteristic case. As an application, one gets that the limits of the physical viscous approximation ∂tU ε ˆ ε
We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space... more We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions to the Cauchy problem for any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions admit time-asymptotic flocking.
Quarterly of Applied Mathematics, 2020
General hyperbolic systems of balance laws with inhomogeneity in space and time in all constituti... more General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.
Calculus of Variations and Partial Differential Equations, 2020
We propose a variational scheme for the construction of isentropic processes of the equations of ... more We propose a variational scheme for the construction of isentropic processes of the equations of adiabatic thermoelasticity with polyconvex internal energy. The scheme hinges on the embedding of the equations of adiabatic polyconvex thermoelasticity into a symmetrizable hyperbolic system. We establish existence of minimizers for an associated minimization Theorem and construct measure-valued solutions that dissipate the total energy. We prove that the scheme converges when the limiting solution is smooth.
Discrete & Continuous Dynamical Systems - A, 2019
For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measur... more For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. However, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure-valued weak versus strong uniqueness using the averaged relative entropy inequality.
Communications in Partial Differential Equations, 2018
We embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hype... more We embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system and derive a relative entropy identity in the extended variables. Following the relative entropy formulation, we prove the convergence from thermoviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero. Also, convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. Finally, we establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.
Archive for Rational Mechanics and Analysis, 2017
We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperb... more We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an L p framework. It also provides a weak-strong uniqueness theorem for measure valued solutions of the hyperbolic problem. In the second part, the relative entropy identity is developed for the systems of gas dynamics for viscous and heat conducting gases and for the system of thermoviscoelasticity both including viscosity and heat-conduction effects. The dissipation mechanisms and the concentration measures play different roles when applying the method to the general class of hyperbolic-parabolic systems and to the specific examples, and their ramifications are highlighted.
On Hyperbolic Balance Laws and Applications
Springer INdAM Series, 2017
An overview of the current state of the theory of general strictly hyperbolic systems of balance ... more An overview of the current state of the theory of general strictly hyperbolic systems of balance laws in one space dimension is documented in this article. Results on global existence, stability and uniqueness of entropy weak solutions are stated and properties such as the decay of positive waves and the rate of convergence of viscous approximations are presented. The article concludes with an application on the existence of non-smooth isometric immersions into \(\mathbb{R}^{3}\).
On the decay rate of the Gauss curvature for isometric immersions
Bulletin of the Brazilian Mathematical Society, New Series, 2016
We address the problem of global embedding of a two dimensional Riemannian manifold with negative... more We address the problem of global embedding of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. A theorem of Efimov states that if the curvature decays too slowly to zero then global smooth immersion is impossible. On the other hand a theorem of J.-X. Hong shows that if decay is sufficiently rapid (roughly like t−(2+δ) for δ > 0) then global smooth immersion can be accomplished. Here we present recent results on applying the method of compensated compactness to achieve a non-smooth global immersion with rough data and we give an emphasis on the role of decay rate of the Gauss curvature.
Journal of Hyperbolic Differential Equations, 2015
General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global... more General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small BV data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.
Uploads
Papers by Cleopatra Christoforou