Papers by Konstantina Trivisa
A. We establish the local well-posedness result for the Cauch y problem of a ghost e ffect... more A. We establish the local well-posedness result for the Cauch y problem of a ghost e ffect system from gas dynamics that derives from kinetic theory. We show t hat his system has a unique classical solution for a finite time for all initial data whose deviatio ns from nonzero background values lie in Sobolev spaces of su fficiently high order and such that its initial temperature is p o itive everywhere.
Journal of Evolution Equations, 2017
We investigate the well posedness of the stochastic Navier-Stokes equations for viscous, compress... more We investigate the well posedness of the stochastic Navier-Stokes equations for viscous, compressible, non-isentropic fluids. The global existence of finite-energy weak martingale solutions for large initial data within a bounded domain of R d is established under the condition that the adiabatic exponent γ > d/2. The flow is driven by a stochastic forcing of multiplicative type, white in time and colored in space. This work extends recent results on the isentropic case, the main contribution being to address the issues which arise from coupling with the temperature equation. The notion of solution and corresponding compactness analysis can be viewed as a stochastic counterpart to the work of Feireisl (Dynamics of viscous compressible fluids, vol 26. Oxford University Press, Oxford, 2004). By including the noise, system (1.1) becomes a model for a turbulent compressible flow, with variable density and temperature. The mathematical study of the compressible stochastic Navier-Stokes equations has seen several recent developments. In the barotropic case, the first result is due to [8], which used a deterministic approach in the case where the coefficients σ k are independent of the fluid variables. In this setting, one can make a convenient change of variables which turns the SPDE into a random PDE. In a more general setting, a stochastic approach is required in order to give a meaning to the noise. Three sets of authors [3,14,17] studied independently the case of more general noise coefficients, establishing the existence of global martingale solutions to (1.1) in the barotropic regime. It should be noted that the article [3] of Breit/Hofmanova was the first to appear on ArXiv, while the article [17] of D. Wang/H. Wang was the first to appear in print. The main focus of the present article is to extend the results of [3,14,17] to the more general setting of (1.1), which allows for changes in the temperature of the fluid. We use a combination of the arguments in [7,14] to treat the stochastic and deterministic sides of the problem, respectively. In particular, we follow the work of E. Feireisl [7] for the notion of solution to the temperature equation. We also adapt various compactness arguments from [7], modified appropriately to the frame of arguments in [14] for building weak martingale solutions. We now turn to a more precise statement of our results.
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consigl... more Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal
Nonlinearity, Sep 28, 2021
In [JT22], the authors proved the existence of local-in-time weak solutions to a model of superfl... more In [JT22], the authors proved the existence of local-in-time weak solutions to a model of superfluidity. The system of governing equations was derived in [Pit59] and couples the nonlinear Schrödinger equation (NLS) and the Navier-Stokes equations (NSE). In this article, we prove a weakstrong type uniqueness theorem for these weak solutions. Only some of their regularity properties are used, allowing room for improved existence theorems in the future, with compatible uniqueness results.
Oberwolfach Reports, 2022
Polymeric fluids arise in many practical applications in biotechnology, medicine, chemistry, indu... more Polymeric fluids arise in many practical applications in biotechnology, medicine, chemistry, industrial processes and atmospheric sciences. In this article, the Doi model for the suspensions of rod-like molecules in a compressible fluid is investigated. The model under consideration couples a Fokker-Planck-type equation on the sphere for the orientation distribution of the rods to the Navier-Stokes equations for compressible fluids, which are now enhanced by additional stresses reflecting the orientation of the rods on the molecular level. The coupled problem is 5-dimensional (three-dimensions in physical space and two degrees of freedom on the sphere) and it describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained. Prescribing arbitrarily the initial density of the fluid, the initial velocity, and the initial orientation distribution in suitable spaces we establish the global-in-time existence of a weak solution to our model defined on a bounded domain in the three dimensional space. The proof relies on the construction of a sequence of approximate problems by introducing appropriate regularization and the establishment of compactness.
SIAM Journal on Numerical Analysis, 2018
We investigate the evolution of tumor growth relying on a nonlinear model of partial differential... more We investigate the evolution of tumor growth relying on a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the role of nutrient and drug application in the progression of tumors. We construct an explicit convergent numerical scheme to approximate solutions of the nonlinear system of partial differential equations. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The same numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug both shrinkage and growth of tumors can occur. The role of inhomogeneous boundary conditions, vascularization and anisotropies in the development of tumor shape irregularities are discussed.
Journal of Nonlinear Science, 2020
The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations are a macroscopic model of superfluidity ... more The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations are a macroscopic model of superfluidity at non-zero temperatures. For smooth, compactly supported data, we prove the global well-posedness of strong solutions to these equations in R 2 , in the incompressible and isothermal case. The proof utilises a contraction mapping argument to establish local well-posedness for high-regularity data, following which we demonstrate global regularity using an analogue of the Beale-Kato-Majda criterion in this context. In the appendix, we address the sufficient conditions on a 2D vorticity field, in order to have a finite kinetic energy.
arXiv: Statistics Theory, 2018
If two probability density functions (PDFs) have values for their first $n$ moments which are qui... more If two probability density functions (PDFs) have values for their first $n$ moments which are quite close to each other (upper bounds of their differences are known), can it be expected that the PDFs themselves are very similar? Shown below is an algorithm to quantitatively estimate this "similarity" between the given PDFs, depending on how many moments one has information about. This method involves the concept of functions behaving "similarly" at certain "length scales", which is also precisely defined. This technique could find use in data analysis, to compare a data set with a PDF or another data set, without having to fit a functional form to the data.
arXiv: Analysis of PDEs, 2018
This article establishes the global existence of weak solutions to a model proposed by Rosensweig... more This article establishes the global existence of weak solutions to a model proposed by Rosensweig (Rosensweig, Ferrohydrodynamics (1985)) for the dynamics of ferrofluids. The system is expressed by the conservation of linear momentum, the incompressibility condition, the conservation of angular momentum, and the evolution of the magnetization. The existence proof is inspired by the DiPerna-Lions theory of renormalized solutions. In addition, the rigorous relaxation limit of the equations of ferrohydrodynamics towards the quasi-equilibrium is investigated. The proof relies on the relative entropy method, which involves constructing a suitable functional, analyzing its time evolution and obtaining convergence results for the sequence of approximating solutions.
We investigate a free boundary problem modeling the growth of tumors cells. The model is given by... more We investigate a free boundary problem modeling the growth of tumors cells. The model is given by a multi-phase flow and the tumor is described as a growing continuum Ω with boundary ∂Ω both of which evolve in time. The model is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Brinkman’s equation. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data. ∗Speaker sciencesconf.org:equadiff2015:71701
Journal of Mathematical Fluid Mechanics, 2022
In [Pit59], a micro-scale model of superfluidity was derived from first principles, to describe t... more In [Pit59], a micro-scale model of superfluidity was derived from first principles, to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model couples two of the most fundamental PDEs in mathematics: the nonlinear Schrödinger equation (NLS) and the Navier-Stokes equations (NSE). In this article, we show the local existence of weak solutions to this system (in a smooth bounded domain in 3D), by deriving the required a priori estimates. (We will also establish an energy inequality obeyed by the weak solutions constructed in [Kim87] for the incompressible, inhomogeneous NSE.) To the best of our knowledge, this is the first rigorous mathematical analysis of a bidirectionally coupled system of the NLS and NSE.
A model is introduced for the dynamic combustion of exothermically reacting, compressible fluids ... more A model is introduced for the dynamic combustion of exothermically reacting, compressible fluids formulated by the Navier-Stokes equations expressing the conservation of mass, the balance of momentum and energy and the two species chemical kinetics. The global existence of the discontinuous solutions to the Navier- Stokes equations with large discontinuous initial data is established by combining the difference approximations techniques and energy estimates. The asymptotic analysis of solutions is also discussed.
Bulletin of the American Physical Society, 2020
Proceedings of the National Academy of Sciences, 2021
Significance Nonlinear differential equations appear in many domains and are notoriously difficul... more Significance Nonlinear differential equations appear in many domains and are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear differential equations have complexity exponential in the evolution time, we give the first quantum algorithm for dissipative nonlinear differential equations that is efficient provided the dissipation is sufficiently strong relative to nonlinear and forcing terms and the solution does not decay too rapidly. We also establish a lower bound showing that differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics. Furthermore, numerical results for the Burgers equation suggest that our algorithm may potentially address complex nonlinear phenomena even in regimes with weaker dissipation.
Kinetic & Related Models, 2020
We investigate the global existence of weak solutions to a free boundary problem governing the ev... more We investigate the global existence of weak solutions to a free boundary problem governing the evolution of finitely extensible bead-spring chains in dilute polymers. The free boundary in the present context is defined with regard to a density threshold of ρ = 1, below which the fluid is modeled as compressible and above which the fluid is modeled as incompressible. The present article focuses on the physically relevant case in which the viscosity coefficients present in the system depend on the polymer number density, extending the earlier work [8]. We construct the weak solutions of the free boundary problem by performing the asymptotic limit as the adiabatic exponent γ goes to ∞ for the macroscopic model introduced by Feireisl, Lu and Süli in [10] (see also [6]). The weak sequential stability of the family of dissipative (finite energy) weak solutions to the free boundary problem is also established.
Oberwolfach Reports, 2020
Dynamics of Partial Differential Equations, 2018
We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. Th... more We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Darcy's equation. The domain occupied by the tumor in this setting is a growing continuum Ω with boundary ∂Ω both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on the vanishing viscosity of the Brinkman's regularization. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data.
Journal of Mathematical Analysis and Applications, 2019
We investigate the global existence of weak solutions to a free boundary problem governing the ev... more We investigate the global existence of weak solutions to a free boundary problem governing the evolution of finitely extensible bead-spring chains in dilute polymers. We construct weak solutions of the two-phase model by performing the asymptotic limit as the adiabatic exponent γ goes to ∞ for a macroscopic model which arises from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids. In this context the polymeric molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. This class of models involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain Ω in R d , d = 2, 3 coupled with a Fokker-Planck-Smoluchowski-type diffusion equation (cf. Barrett and Süli [4],[5], [9]). The convergence of these solutions, up to a subsequence, to the free-boundary problem is established using weak convergence methods, compactness arguments which rely on the monotonicity properties of certain quantities in the spirit of [19].
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Papers by Konstantina Trivisa