We define two nonlinear shell models whereby the deformation of an elastic shell with small thickness minimizes ad hoc functionals over sets of admissible deformations of Kirchhoff-Love type. We establish that both models are close in a specific sense to the well-known nonlinear shell model of W.T. Koiter and that one of them has a solution, by contrast with Koiter's model for which such an existence theorem is yet to be proven.
We consider a previously proposed general nonlinear poromechanical formulation, and we derive a linearized version of this model. For this linearized model, we obtain an existence result and we propose a complete discretization strategy-in time and space-with a special concern for issues associated with incompressible or nearly-incompressible behavior. We provide a detailed mathematical analysis of this strategy, the main result being an error estimate uniform with respect to the compressibility parameter. We then illustrate our approach with detailed simulation results and we numerically investigate the importance of the assumptions made in the analysis, including the fulfillment of specific inf-sup conditions.
The advection-diffusion equation y_{t}^{ε} -εy_{xx}^{ε} + My_{x}^{ε}=0, (x, t) ∈ (0, 1)×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient ε goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to ε if T is greater than a minimal time T_{M}, with T_{M} in the interval[1, 2√3]/M for M > 0 and in the interval[2√2, 2(1 + √3)]/|M|for M < 0. The exact value of T_{M} is however unknown. We investigate in this work the determination of the minimal time T_{M} employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and ε. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter ε, of the optimality system associated to the control of minimal L^{2}-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.
The human tricuspid valve, one of the key cardiac structures, plays an important role in the circulatory system. However, there are few mathematical models to accurately simulate it. In this paper, firstly, we consider the tricuspid valve as an elastic shell with a specific shape and establish its novel geometric model. Concretely, the anterior, the posterior and the septal leaflets of the valve are supposed to be portions of the union of two interfacing semi-elliptic cylindrical shells when they are fully open. Next, we use Koiter's linear shell model to describe the tricuspid valve leaflets in the static case, and provide a numerical scheme for this elastostatics model. Specifically, we discretize the space variable, i.e., the two tangent components of the displacement are discretized by using conforming finite elements (linear triangles) and the normal component of the displacement is discretized by using conforming Hsieh-Clough-Tocher triangles (HCT triangles). Finally, we make numerical experiments for the tricuspid valve and analyze the outcome. The numerical results show that the proposed mathematical model describes well the human tricuspid valve subjected to applied forces.
This paper is concerned with the inflow problem for one-dimensional compressible Navier-Stokes equations. For such a problem, Huang, Matsumura, and Shi showed in[4] that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable provided that both the initial perturbation and the boundary velocity are assumed to be sufficiently small. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and our main idea is to use the smallness of the strength of the viscous shock wave and the boundary velocity to control the possible growth of the solutions induced by the nonlinearity of the compressible Navier-Stokes equations and the inflow boundary condition. The key point in our analysis is to deduce the desired uniform positive lower and upper bounds on the density.
A formal asymptotics leading from a system of Boltzmann equations for mixtures towards either Vlasov-Navier-Stokes or Vlasov-Stokes equations of incompressible fluids was established by the same authors and Etienne Bernard in:A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory Commun. Math. Sci., 15:1703-1741 (2017) and A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures. KRM, 11:43-69 (2018). With the same starting point but with a different scaling, we establish here a formal asymptotics leading to the Vlasov-Euler system of compressible fluids. Explicit formulas for the coupling terms are obtained in two typical situations:for elastic hard spheres on one hand, and for collisions corresponding to the inelastic interaction with a macroscopic dust speck on the other hand.
This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting.
In this paper, we consider the lifespan of solution to the MHD boundary layer system as an analytic perturbation of general shear flow. By using the cancellation mechanism in the system observed in[12], the lifespan of solution is shown to have a lower bound in the order of ε^{-2-} if the strength of the perturbation is of the order of ε. Since there is no restriction on the strength of the shear flow and the lifespan estimate is larger than the one obtained for the classical Prandtl system in this setting, it reveals the stabilizing effect of the magnetic field on the electrically conducting fluid near the boundary.
This paper surveys some results on the existence and stability of solutions to some partial differential equations of gaseous stars in the framework of Newtonian mechanics, and presents some key ideas in the proofs.