Special Session 15: On the dynamics of hyperbolic partial differential equations: theory and applications

Bernstein`s Problem and Positivity Preserving Exponential Integrators for Evolution Equations

Rachid Ait Haddou King Fahd University of Petroleum and Minerals Saudi Arabia Co-Author(s):    Huda Altamimi

Abstract: Bernstein`s problem asks for the maximum positive real number $R$ for which an absolutely monotonic function, with a specified number of derivatives at the origin, exists on the interval $(-R,0)$. Optimal threshold factors govern the maximum allowable step-size for positivity preserving integration methods of initial-value problems. This talk establishes a link between Bernstein`s problem and optimal threshold factors and presents algorithms for computing the latter. Derivation of optimal exponential integrators of specified accuracy for evolution equations is discussed.

Asymptotic Limits for Strain-Gradient Viscoelasticity with Nonconvex Energy

Aseel AlNajjar King Abdullah University of Science and Technology Saudi Arabia Co-Author(s):    Stefano Spirito and Athanasios E. Tzavaras

Abstract: We consider the system of viscoelasticity with higher-order gradients and nonconvex energy in several space dimensions. After deriving the compactness estimates for the system, we first establish global existence of weak solutions. We then study the asymptotic limits as the viscosity tends to zero or as the coefficient of the higher-order gradient vanishes. In the latter problem and for the two-dimensional case, we also prove a stability result for the solutions in the regularity class and establish a rate of convergence.

Euler-Riesz systems: Compensated Integrability and connections to Harmonic Analysis

Nuno J Alves University of Vienna Austria Co-Author(s):    Loukas Grafakos and Athanasios E. Tzavaras

Abstract: In this talk, we consider a Euler-Riesz system and establish a higher integrability estimate for the density. This is achieved through a reformulation of the system and compensated integrability. We will also discuss connections to harmonic analysis, particularly the introduction of a bilinear fractional integral operator, whose uniform estimates are derived via restricted weak-type endpoint bounds and Marcinkiewicz interpolation.

A phase separation model for binary fluids with hereditary viscosity

Maurizio Grasselli Politecnico di Milano Italy Co-Author(s):    Andrea Poiatti

Abstract: We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier-Stokes-Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term incorporating hereditary effects. These equations are coupled with the Cahn-Hilliard equation with Flory-Huggins type potential. The resulting system is subject to no-slip condition for the (volume averaged) fluid velocity and no-flux boundary conditions for the order parameter as well as for the chemical potential. The presence of a memory term entails hyperbolic features (i.e. the fluid velocity does not regularize in finite time). The corresponding initial and boundary value problem is well posed. Moreover, by adding an Eckman-type damping, we show that it defines a dissipative dynamical system in a suitable phase space, i.e., there is a bounded absorbing set. Then, we discuss the existence of global and exponential attractors.

On the well-posedness and stability for carbon nanotubes as coupled two Timoshenko beams with frictional dampings

Aissa Guesmia Lorraine University France Co-Author(s):

Abstract: The objective of this work is to study the well-posedness and stability questions for double wall carbon nanotubes modeled as linear one-dimensional coupled two Timoshenko beams in a bounded domain under frictional dampings. First, we prove the well-posedness of our system by applying the semigroups theory of linear operators. Second, we show several strong, non-exponential, exponential, polynomial and non-polynomial stability results depending on the number of frictional dampings, their position and some connections between the coefficients. In some cases, the optimality of the polynomial decay rate is also proved. The proofs of these stability results are based on a combination of the energy method and the frequency domain approach. For the details, see the following paper: A. Guesmia, On the well-posedness and stability for carbon nanotubes as coupled two Timoshenko beams with frictional dampings, J. Appl. Anal. Comp., 14 (2024), 1-50.

On the Cauchy problem of the MGT-Viscoelastic plate with heat conduction of Fourier law

Bounadja Hizia University of Sciences and Technology Houari Boumediene Algeria Co-Author(s):    Mounir Afilal & Abdelaziz Soufyane

Abstract: In this paper, we consider the MGT viscoelastic plate model in $\mathbb{R}^{N}$ with heat conduction of Fourier law. First, we give the appropriate functional setting needed for the well-posedness. Second, using the energy method in the Fourier space, we obtain the decay rate of a norm related to the solutions in both cases sub-critical and critical cases. In particular, we prove that the decay rate does not exhibit the well-known regularity loss phenomenon present in some models existing in the literature.

A new duality method for mean-field limits with singular interactions

Pierre-Emmanuel Jabin Pennsylvania State University USA Co-Author(s):    D. Bresch, M. Duerinckx

Abstract: We introduce a new approach to justify mean-field limits for first- and second-order particle systems with singular interactions. It is based on a duality approach combined with the analysis of linearized dual correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces with a possibly vanishing temperature parameter.

On a truncated thermoelastic Timoshenko System with a dual-phase-lag model

Salim Messaoudi University of Sharjah United Arab Emirates Co-Author(s):    Ahmed keddi and Mohamed Alahyane

Abstract: In this work, we consider a one-dimensional truncated Timoshenko system coupled with a heat equation, where the heat flux is given by the generalized dual-phase lag model. By using the semigroup theory and some non classical diferential operators, we establish the well-posedness of the problem. Then, we use the multiplier method to show that the only one heat control is enough to stabilize the whole system exponentially without imposing the usual equal-speed assumption or any other stability number. Moreover, to illustrate our theoretical results, we give some numerical tests. Our result seems to be the first of this type.

The Westervelt-Pennes model of nonlinear thermo-acoustics: local well-posedness and singular limit for vanishing relaxation time

Belkacem Said-Houari University of Sharjah United Arab Emirates Co-Author(s):

Abstract: In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt-Pennes-Cattaneo model). Using the energy method together with a fixed point argument, we prove that our model is locally well-posed and does not degenerate under a smallness assumption on the pressure data in the Westervelt equation. In addition, we perform a singular limit analysis and show that the Westervelt-Pennes-Fourier model can be seen as an approximation of the Westervelt-Pennes-Cattaneo model as the relaxation parameter tends to zero. This is done by deriving uniform bounds of the solution with respect to the relaxation parameter.

STATIONARY SHEAR FLOW OF NEMATIC LIQUID CRYSTALS: MULTIPLICITY, STABILITY, AND BIFURCATION

Majed Sofiani King Abdullah University of Science and Technology (KAUST) Saudi Arabia Co-Author(s):    Weishi Liu

Abstract: Liquid crystal is a state of matter that has crystal properties and can flow like a liquid. These two characteristics interact with each other so that any distortion of the crystal structure affects the flow, and vice versa. In this talk, I will present recent results on the boundary value problem for the shear flow of nematic liquid crystals via the parabolic Ericksen-Leslie model. In particular, I will discuss the existence of multiple stationary solutions, bifurcation phenomenon, and stability/instability of bifurcating solutions.

Cahn-Hillard and Keller-Segel systems as high-friction limits of gas dynamics

Agnieszka Swierczewska-Gwiazda University of Warsaw Poland Co-Author(s):

Abstract: Several recent studies considered the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation using the relative entropy method. Applying the recent result about relations between non-local and local Cahn-Hilliard, we also derive rigorously the large-friction nonlocal- to-local limit. The result is formulated for dissipative measure-valued solutions of the nonlocal Euler-Korteweg equation which are known to exist on arbitrary intervals of time. This approach provides a new method to derive equations not enjoying classical solutions via relative entropy method by introducing the nonlocal effect in the fluid equation. During the talk I will also discuss the high-friction limit of the Euler-Poisson system.

Evolution problems with non-small amplitudes

NAASER-EDDINE TATAR KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS Saudi Arabia Co-Author(s):    Mohamed Kafini

Abstract: In most of the existing works on partial differential equations of hyperbolic type, vibrations or oscillations are assumed to be small in a certain sense. Whilst this is acceptable in many situations, it may be considered as a severe simplification in other situations. Indeed, the obtained simpified models are much simpler but may be far away from the reality in case of large amplitudes. In this presentation, we will briefly discuss some evolution problems with large amplitudes. The main difficulties will be highlighted and some new elements will be put forward.

Sustained Oscillations in Hyperbolic-Parabolic Systems

Athanasios Tzavaras King Abdullah University of Science and Technology Saudi Arabia Co-Author(s):

Abstract: We provide examples of sustained oscillations for hyperbolic-parabolic systems. The existence of sustained oscillations in hyperbolic-parabolic system is studied systematically via examples, for paradigm systems from viscoelasticity and for the compressible Navier-Stokes system with non-monotone pressures. In several space dimensions oscillatory examples are associated with lack of rank-one convexity of the stored energy. The subject naturally leads to the problem of deriving effective equations for the associated homogenization problems. We address that in a context of a simple one-dimensional model be addressed by employing ideas from the kinetic formulation for conservation laws.

The wave equation with acoustic boundary conditions on non-locally reacting surfaces

Enzo Vitillaro Universit\\`a degli Studi di Perugia Italy Co-Author(s):    Delio Mugnolo

Abstract: We deal with the wave equation in a suitably regular open domain of the Euclidean space, supplied with an acoustic boundary condition on a part of the boundary and a homogeneous Neumann boundary condition on the (possibly empty) remaining part of it. This problem has been proposed a long time ago by Beale and Rosencrans, to model acoustic wave propagation with locally reacting boundary, and it has been the object of a wide literature. The case of non--locally reacting boundaries, when the homogeneous Neumann boundary condition is replaced by the mathematically more attracting homogeneous Dirichlet boundary condition, has been studied as well. The physical derivation of the problem is treated in the talk by the author in SS96. In this talk, we focus on non-locally reacting boundaries without any Dirichlet boundary condition. We first give well-posedness results in the natural energy space and regularity results. Hence, we shall give precise qualitative results for solutions when the domain is bounded and sufficiently regular. The results presented will appear in the Memoirs of the American Mathematical Society and are available at the address https://doi.org/10.48550/arXiv.2105.09219.

Energy methods for an improved blow-up bound for a superconformal wave equation

Hatem Zaag CNRS and Universite Sorbonne Paris Nord France Co-Author(s):    M.A. Hamza

Abstract: We address the blow-up rate issue for the Nonlinear Wave Equation (NLW) with superlinear power nonlinearity. For subconformal and conformal power, the blow-up rate is given by the solution of the associated ODE, as this was shown by Merle and Zaag in 2003 and 2005. In the superconformal case below the Sobolev exponent, various bounds are known, from the work of Killip, Stovall and Visan in 2014, and also in our earlier paper in 2013. The aim of this talk is to give a better bound. Our method relies on some energy estimates in similarity variables, where we consider the superconformal case as a (large) perturbation of the conformal case. This leads to some exponential bound on the self-similar version, directly related to the size of the large perturbation.