A first-order-accurate-in-time, finite difference scheme is proposed and analyzed for the Ericksen–Leslie system, which describes the evolution of nematic liquid crystals. For the penalty function to approximate the constraint $\lvert d \rvert = 1$, a convex-concave decomposition for the corresponding energy functional is applied. In addition, appropriate semi-implicit treatments are adopted for the convection terms, for both the velocity vector and orientation vector, as well as the coupled elastic stress terms. In turn, all the semi-implicit terms can be represented as a linear operator of a vector potential, and its combination with the convex splitting discretization for the penalty function leads to a unique solvability analysis for the proposed numerical scheme. Furthermore, a careful estimate reveals an unconditional energy stability of the numerical system, composed of the kinematic energy and internal elastic energies. More importantly, we provide an optimal rate convergence analysis and error estimate for the numerical scheme. In addition, a nonlinear iteration solver is outlined, and the numerical accuracy test results are presented, which confirm the optimal rate convergence estimate.
In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists of a bi-species Vlasov–Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the two Vlasov equations and a strong solution for the Poisson equation. The first parts of the article are devoted to explaining the model and proceed to a detailed study of the Vlasov equations. This study then leads to a reformulation of the Poisson equation as a 1D non-linear and non-local equation and we prove it admits a strong solution using an iterative fixed-point procedure.
In this work, we study the following Neumann-initial boundary value problem for a three-component chemotaxis model describing tumor angiogenesis:
\[
\left \{
\begin{array}
u_t = \Delta u-\chi \nabla \cdot (u \nabla v)+ \xi_1 \nabla \cdot (u \nabla w)+u(a-\mu u^\theta ), & x \in \Omega , t \gt 0, \\
v_t = d \Delta v+ \xi_2 \nabla \cdot (v \nabla w)+u-v, & x \in \Omega , t \gt 0, \\
0= \Delta w+u-\overline{u}, \int_{\Omega} w=0, \overline{u}:= \frac{1}{\lvert\Omega\vert} \int_{\Omega} u, & x \in \Omega , t \gt 0, \\
\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} =0, & x \in \partial \Omega , t \gt 0, \\
u(x, 0)=u_0(x), v(x, 0)=v_0(x), & x \in \Omega ,
\end{array}
\right .
\]
in a bounded smooth but not necessarily convex domain $\Omega \subset \mathbb{R}^n (n \geq 2)$ with model parameters $\chi_1, \chi_2, d, \theta \gt 0, a, \chi , \mu \geq 0$. Based on subtle energy estimates, we first identify two positive constants $\chi_0$ and $\mu_0$ such that the above problem allows only global classical solutions with qualitative bounds provided one of the following conditions holds:
Then, due to the obtained qualitative bounds, upon deriving higher order gradient estimates, we show exponential convergence of bounded solutions to the spatially homogeneous equilibrium (i) for $\mu$ large if $\mu \gt 0$, (ii) for $d$ large if $a=\mu=0$ and (iii) for merely $d \gt 0$ if $\chi=a=\mu =0$. As a direct consequence of our findings, all solutions to the above system with $\chi=a=\mu=0$ are globally bounded and they converge to constant equilibrium, and therefore, no patterns can arise.
This article is concerned with the Cauchy problem to a multi-dimensional two-fluid plasma model in critical functional framework which is not related to the energy space. When the initial data are close to a stable equilibrium state in the sense of suitable $L^p$-type Besov norms, the global well-posedness for the multi-dimensional system is shown. As a consequence, one may exhibit the unique global solution for a class of large highly oscillating initial velocities in physical dimensions $N=2,3$. Furthermore, based on refined time weighted inequalities in the Fourier spaces, we also establish optimal large-time behavior for the constructed global solutions under a mild additional decay assumption involving only the low frequencies of the initial data.
This paper studies a residual-based a posteriori error estimator for partially penalized immersed finite element (PPIFE) approximation to elliptic interface problems. Utilizing the error equation for the PPIFE approximation, we construct an a posteriori error estimator. Properly weighted coefficients are proposed for the terms in indicators to overcome the dependence of the efficiency constants on the jump of the diffusion coefficients across the interface. The PPIFE method is based on non-body-fitted mesh, and hence we perform detailed analysis on the local efficiency bounds of the estimator on regular and irregular interface elements with different techniques. We introduce a new approach, which does not involve the Helmholtz decomposition, to give the reliability bounds of the estimator with an $L^2$ representation of the true error as the main tool. More importantly, the efficiency and reliability constants are independent of the interface location and the mesh size. Numerical experiments are provided to illustrate the efficiency of the estimator and the adaptive mesh refinement for different jump rates or interface geometries.
This article is devoted to the analysis of quasilinear degenerate chemotaxis system with two-species in dimension $d \geq 3$. The global existence of weak solution to the chemotaxis system with small initial data is proved for the super-critical case in both parabolic-elliptic and fully parabolic types.
In this article we prove the global existence of a unique strong solution to the initial boundary-value problem for a fourth-order exponential PDE. The equation we study was originally proposed to study the evolution of crystal surfaces, and was derived by applying a nonstandard scaling regime to a microscopic Markov jump process with Metropolis rates. Our investigation here finds that compared to the PDEs which use Arrhenius rates (and also have a fourth-order exponential nonlinearity), the hyperbolic sine nonlinearity in our equation can offer much better control over the exponent term even in high dimensions.
We extend a method [E. Cancès and L.R. Scott, SIAM J. Math. Anal., 50:381–410, 2018] to compute more terms in the asymptotic expansion of the van der Waals attraction between two hydrogen atoms. These terms are obtained by solving a set of modified Slater–Kirkwood partial differential equations. The accuracy of the method is demonstrated by numerical simulations and comparison with other methods from the literature. It is also shown that the scattering states of the hydrogen atom, that are the states associated with the continuous spectrum of the Hamiltonian, have a major contribution to the $\mathrm{C}_6$ coefficient of the van der Waals expansion.
We present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and $C^2$ approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H‑Theorem. We construct convex approximations to the Maxwell–Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.
We present a general, constructive method to derive thermodynamically consistent models and consistent dynamic boundary conditions hierarchically following the generalized Onsager principle. The method consists of two steps in tandem: the dynamical equation is determined by the generalized Onsager principle in the bulk firstly, and then the surface chemical potential and the thermodynamically consistent boundary conditions are formulated subsequently by applying the generalized Onsager principle at the boundary. The application strategy of the generalized Onsager principle in two steps yields thermodynamically consistent models together with the consistent boundary conditions that warrant a non-negative entropy production rate (or equivalently non-positive energy dissipation rate in isothermal cases) in the bulk as well as at the boundary. We illustrate the method using phase field models of binary materials elaborated on two sets of thermodynamically consistent dynamic boundary conditions. These two types of boundary conditions differ in how the across boundary mass flux participates in the surface dynamics at the boundary. We then show that many existing thermodynamically consistent, binary phase field models together with their dynamic or static boundary conditions are derivable from this approach. As an illustration, we show numerically how dynamic boundary conditions affect crystal growth in the bulk using a binary phase field model.