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Discrete & Continuous Dynamical Systems - B

October 2020 , Volume 25 , Issue 10

Special issue on PDEs and their applications at DEA 2019

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Yoshikazu Giga, Peter Kloeden, Irena Lasiecka, Peter Markowich, Elisabetta Rocca, Enrico Valdinoci, Enrique Zuazua and Krzysztof Ciepliński
2020, 25(10): i-ii doi: 10.3934/dcdsb.2020265 +[Abstract](150) +[HTML](68) +[PDF](78.88KB)
Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity
Samuel Amstutz and Nicolas Van Goethem
2020, 25(10): 3769-3805 doi: 10.3934/dcdsb.2020240 +[Abstract](234) +[HTML](82) +[PDF](779.47KB)

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of the Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.

Higher-order time-stepping schemes for fluid-structure interaction problems
Daniele Boffi, Lucia Gastaldi and Sebastian Wolf
2020, 25(10): 3807-3830 doi: 10.3934/dcdsb.2020229 +[Abstract](276) +[HTML](96) +[PDF](2988.16KB)

We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of second order based on backward differentiation formulae and on the Crank–Nicolson method. We show the stability properties of the resulting method; numerical tests confirm the theoretical results.

Generalized solutions to models of inviscid fluids
Dominic Breit, Eduard Feireisl and Martina Hofmanová
2020, 25(10): 3831-3842 doi: 10.3934/dcdsb.2020079 +[Abstract](628) +[HTML](334) +[PDF](434.83KB)

We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible oscillations and/or concentrations in the associated generating sequence. Unlike the conventional measure–valued solutions or rather their expected values, the dissipative solutions comply with a natural compatibility condition – they are classical solutions as long as they enjoy a certain degree of smoothness.

Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth
Sun-Sig Byun, Yumi Cho and Shuang Liang
2020, 25(10): 3843-3855 doi: 10.3934/dcdsb.2020038 +[Abstract](676) +[HTML](317) +[PDF](440.73KB)

Quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth are studied. We obtain a global Calderón-Zygmund estimate for such an irregular obstacle problem by proving that the gradient of the solution is as integrable as both the nonhomogeneous term and the gradient of the associated double obstacles under minimal regularity requirements on the elliptic operator over the boundary of the nonsmooth domain.

Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media
Yves Capdeboscq and Haun Chen Yang Ong
2020, 25(10): 3857-3887 doi: 10.3934/dcdsb.2020228 +[Abstract](187) +[HTML](76) +[PDF](521.67KB)

We consider the conductivity equation in a bounded domain in \begin{document}$ \mathbb{R}^{d} $\end{document} with \begin{document}$ d\geq3 $\end{document}. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions \begin{document}$ DU $\end{document} takes both positive and negatives values. In this work, we construct a class of inclusions \begin{document}$ Q $\end{document} and boundary conditions \begin{document}$ \phi $\end{document} such that the determinant of the solution of the boundary value problem satisfies this sign-changing constraint. We provide lower bounds for the measure of the sets where the Jacobian determinant is greater than a positive constant (or lower than a negative constant). Different sign changing structures where introduced in [9], where the existence of such media was first established. The quantitative estimates provided here are new.

Exact controllability of the linear Zakharov-Kuznetsov equation
Mo Chen and Lionel Rosier
2020, 25(10): 3889-3916 doi: 10.3934/dcdsb.2020080 +[Abstract](573) +[HTML](309) +[PDF](516.06KB)

We consider the linear Zakharov-Kuznetsov equation on a rectangle with a left Dirichlet boundary control. Using the flatness approach, we prove the null controllability of that equation and provide a space of analytic reachable states.

On the 1D modeling of fluid flowing through a Junction
Rinaldo M. Colombo and Mauro Garavello
2020, 25(10): 3917-3929 doi: 10.3934/dcdsb.2019149 +[Abstract](1477) +[HTML](654) +[PDF](548.43KB)

A compressible fluid flows through a junction between two different pipes. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather costly. This note compares different 1D approaches to this phenomenon.

Estimating the division rate from indirect measurements of single cells
Marie Doumic, Adélaïde Olivier and Lydia Robert
2020, 25(10): 3931-3961 doi: 10.3934/dcdsb.2020078 +[Abstract](562) +[HTML](309) +[PDF](3440.56KB)

Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.

Null controllability of one dimensional degenerate parabolic equations with first order terms
J. Carmelo Flores and Luz De Teresa
2020, 25(10): 3963-3981 doi: 10.3934/dcdsb.2020136 +[Abstract](382) +[HTML](178) +[PDF](417.08KB)

In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.

Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms
Yoshikazu Giga, Hiroyoshi Mitake and Hung V. Tran
2020, 25(10): 3983-3999 doi: 10.3934/dcdsb.2019228 +[Abstract](1302) +[HTML](495) +[PDF](432.93KB)

We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.

Classical Langevin dynamics derived from quantum mechanics
Håkon Hoel and Anders Szepessy
2020, 25(10): 4001-4038 doi: 10.3934/dcdsb.2020135 +[Abstract](383) +[HTML](176) +[PDF](1100.34KB)

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process
Karel Kadlec and Bohdan Maslowski
2020, 25(10): 4039-4055 doi: 10.3934/dcdsb.2020137 +[Abstract](371) +[HTML](187) +[PDF](438.37KB)

An ergodic control problem is studied for controlled linear stochastic equations driven by cylindrical Lévy noise with unbounded control operator in a Hilbert space. A family of optimal controls is shown to consist of those asymptotically achieving the feedback form that employs the corresponding Riccati equation. The formula for optimal cost is given. The general results are applied to stochastic heat equation with boundary control and to stochastic structurally damped plate equations with point control.

Hexagonal spike clusters for some PDE's in 2D
Theodore Kolokolnikov and Juncheng Wei
2020, 25(10): 4057-4070 doi: 10.3934/dcdsb.2020039 +[Abstract](498) +[HTML](286) +[PDF](759.49KB)

We study hexagonal spike cluster patterns for Gierer-Meinhardt reaction-diffusion system with a precursor on all of \begin{document}$ \mathbb R^2 $\end{document}. These clusters consist of \begin{document}$ N $\end{document} spikes which form a nearly hexagonal lattice of a finite size. The lattice density is locally nearly constant, but globally non-uniform. We also characterize a similar hexagonal spike cluster steady state for a simple elliptic PDE \begin{document}$ 0 = \Delta u - u +u^2 + \varepsilon |x|^2 $\end{document} with a small "confinement well" \begin{document}$ \varepsilon |x|^2 $\end{document}. The key idea is to explicitly exploit the local hexagonality structure to asymptotically approximate the solution using certain lattice sums. In the limit of many spikes, we derive the effective spike density as well as the cluster radius. This effective density is a solution to a certain separable first-order ODE coupled to an integral boundary condition.

Uniform stabilization of Boussinesq systems in critical $ \mathbf{L}^q $-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls
Irena Lasiecka, Buddhika Priyasad and Roberto Triggiani
2020, 25(10): 4071-4117 doi: 10.3934/dcdsb.2020187 +[Abstract](841) +[HTML](168) +[PDF](919.86KB)

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will be minimal in number, and of reduced dimension: more precisely, they will be of dimension \begin{document}$ (d-1) $\end{document} for the fluid component and of dimension \begin{document}$ 1 $\end{document} for the heat component. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to \begin{document}$ \mathbf{L}^3(\Omega $\end{document}) for \begin{document}$ d = 3 $\end{document}) and the space \begin{document}$ L^q(\Omega $\end{document}) for the thermal component, \begin{document}$ q > d $\end{document}. Thus, this paper may be viewed as an extension of [63], where the same interior localized uniform stabilization outcome was achieved by use of finite dimensional feedback controls for the Navier-Stokes equations, in the same Besov setting.

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