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In this contribution, we present a modelling and simulation framework for parametrised lithium-ion battery cells. We first derive a continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterised non-linear system of partial differential equations, the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modelling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modelling framework.
We present a mathematical model built to describe the fluid dynamics for the heat transfer fluid in a parabolic trough power plant. Such a power plant consists of a network of tubes for the heat transport fluid. In view of optimisation tasks in the planning and in the operational phase, it is crucial to find a compromise between a very detailed description of many possible physical phenomena and a necessary simplicity needed for a fast and robust computational approach. We present the model, a numerical approach, simulation for single tubes and also for realistic network settings. In addition, we optimise the power output with respect to the operational parameters.
For $N\geq 2$, a bounded smooth domain $\Omega$ in $\mathbb {R}^{N}$, and $g_0,\, V_0 \in L^{1}_{loc}(\Omega )$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem:
\[ -\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g |\phi|^{p-2}\phi \text{ in } \Omega, \quad \phi=0 \text{ on } \partial \Omega, \]
where $g$ and $V$ vary over the rearrangement classes of $g_0$ and $V_0$, respectively. We prove the existence of a minimizing pair $(\underline {g},\,\underline {V})$ and a maximizing pair $(\overline {g},\,\overline {V})$ for $g_0$ and $V_0$ lying in certain Lebesgue spaces. We obtain various qualitative properties such as polarization invariance, Steiner symmetry of the minimizers as well as the associated eigenfunctions for the case $p=2$. For annular domains, we prove that the minimizers and the corresponding eigenfunctions possess the foliated Schwarz symmetry.
In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns.21 (1996), 363–394); Grenier (Commun. Pure Appl. Math.50 (1997), 821–865); Ukai (J. Math. Kyoto Univ.26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.
The boundary element method for the eddy current problem (BEM-ECP) was proposed in a number of papers and is applicable to important tasks such as the problem of inductive heating and transmission of electromagnetic energy. BEM-ECP requires the construction of a system of linear algebraic equations in which the matrix is inherently dense and is constructed out of element matrices. For the process of the element matrix computation, two cases are normally considered: far-field interaction and near-field interaction, because the construction of element matrices requires integration of a singular function. In this article, we suggest a transform that allows computing the matrix components of the near-singular interaction part while implementing only the single and double layer potentials. The previously suggested modified double layer potential (MDLP) can be integrated by means of this transform, which simplifies the program implementation of BEM-ECP significantly. Solving model problems, we analyse the drawbacks of the previously suggested approach. This analysis includes the proof of the MDLP singularity that makes the integration of this potential a rather difficult task without the help of our transform. The previously suggested approach does not work well with surfaces that are not smooth. Our approach does consider such cases, which is its main advantage. We demonstrate this on the model problems with known analytical solutions.
is considered along with no-flux boundary conditions for $u$ and with prescribed constant positive Dirichlet boundary data for $v$. It is shown that if $D\in C^{3}([0,\infty ))$ is such that $0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$ for all $\xi >0$ with some ${K_D}>0$ and $\alpha >0$, then for all initial data from a considerably large set of radial functions on $\Omega$, the corresponding initial-boundary value problem admits a solution blowing up in finite time.
In this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda (Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79). We assume the viscosity of the inclusion fluid is $z\mu _1$ and the viscosity of the hosting fluid is $\mu _1\in \mathbb {R}^{+}$, $z\in \mathbb {C}$. The proof is carried out by the construction of solutions for large $|z|$ and small $|z|$ with an iteration process similar to the one used in Bruno and Leo (Arch. Ration. Mech. Anal. 121 (1993), 303–338) and Golden and Papanicolaou (Commun. Math. Phys. 90 (1983), 473–491) and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (3.17) with different values of contrast parameter $s:=1/(z-1)$, as long as $s$ is outside the interval $\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]$, where the positive constants $E_1$ and $E_2$ are the extension constants that depend only on the geometry of the periodic pore space of the material.
We construct a new type of planar Euler flows with localized vorticity. Let
$\kappa _i\not =0$
,
$i=1,\ldots , m$
, be m arbitrarily fixed constants. For any given nondegenerate critical point
$\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})$
of the Kirchhoff–Routh function defined on
$\Omega ^m$
corresponding to
$(\kappa _1,\ldots , \kappa _m)$
, we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and concentrate on curves perturbed from small circles centered near
$x_{0,i}$
,
$i=1,\ldots ,m$
. The proof is accomplished via the implicit function theorem with suitable choice of function spaces.
The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in $\mathbb {R}^d$, $d\geq 1$, which extends and improves the results obtained by Sideris et al. in ‘T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816’. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the $L^2$-norm of $\nabla u$ is exponential decay.
This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.
Steady-state diffusion in long axisymmetric structures is considered. The goal is to assess one-dimensional approximations by comparing them with axisymmetric eigenfunction expansions. Two problems are considered in detail: a finite tube with one end that is partly absorbing and partly reflecting; and two finite coaxial tubes with different cross-sectional radii joined together abruptly. Both problems may be modelled using effective boundary conditions, containing a parameter known as the trapping rate. We show that trapping rates depend on the lengths of the finite tubes (and that they decay slowly as these lengths increase) and we show how trapping rates are related to blockage coefficients, which are well known in the context of potential flow along tubes of infinite length.
with positive parameters $D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$, $\alpha _u,\alpha _w,\mu _u,\beta$. When posed under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^{2}$, and along with initial conditions involving suitably regular data, the global existence of classical solution to this system was asserted in Tao and Winkler (2020, J. Differ. Equ. 268, 4973–4997). Based on the suitable quasi-Lyapunov functional, it is shown that when the virus replication rate $\beta <1$, the global classical solution $(u,v,w,z)$ is uniformly bounded and exponentially stabilizes to the constant equilibrium $(1, 0, 0, 0)$ in the topology $(L^{\infty }(\Omega ))^{4}$ as $t\rightarrow \infty$.
In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.
This paper investigates the stability of a fully parabolic–parabolic-fluid (PP-fluid) system of the Keller–Segel–Navier–Stokes type in a bounded planar domain under the natural volume-filling hypothesis. In the limit of fast signal diffusion, we first show that the global classical solutions of the PP-fluid system will converge to the solution of the corresponding parabolic–elliptic-fluid (PE-fluid) system. As a by-product, we obtain the global well-posedness of the PE-fluid system for general large initial data. We also establish some new exponential time decay estimates for suitable small initial cell mass, which in particular ensure an improvement of convergence rate on time. To further explore the stability property, we carry out three numerical examples of different types: the nontrivial and trivial equilibriums, and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.
We consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics $-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$ in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms $h(s)$ with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$ when the parameter $\kappa >0$ is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.
We study stationary solutions to the Keller–Segel equation on curved planes. We prove the necessity of the mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case. Furthermore, we provide a correspondence between stationary solutions to the static Keller–Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller–Segel equation (with any mass). Finally, as a complementary result, we prove a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality and use it to show that the Keller–Segel free energy is bounded from below exactly when the mass is $8 \pi$, even in the curved case.
We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.
The long-time behaviour of solutions to the defocussing modified Korteweg-de Vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through
$\overline {\partial }$
-derivatives. To extend the asymptotics to solutions with initial data in lower-regularity spaces, we apply a global approximation via PDE techniques.
We consider the $\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.