Methods and Applications of Analysis

In this paper we study a system of partial differential equations which models lithium-ion batteries. The system describes the conservation of Lithium and conservation of charges in the solid and electrolyte phases, together with the conservation of energy. The mathematical challenge is due to the fact that the reaction terms in the system involve the hyperbolic sine function along with possible degeneracy in one of the high-order terms. We obtain a local existence assertion for the initial boundary problem for the system. In particular, a lower bound for the blow-up time can be derived from our result. We hope that our analysis can lead to a deeper understanding of battery life.

Let $E$ be the attractor of an iterated function system ${\lbrace \phi_i (x) = \rho R_i x + a_i \rbrace}^N_{i=1}$ on $\mathbb{R}^d$, where $0 \lt \rho \lt 1$, $a_i \in \mathbb{R}^d$ and $R_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that ${\lbrace \phi_i \rbrace}^N_{i=1}$ satisfies the open set condition, but not the strong separation condition. We show that $E$ can not be generated by any iterated function system of similitudes satisfying the strong separation condition. This gives a partial answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condition for the generalized Rosenau–Korteweg–de Vries–Burgers equation. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity. We can further obtain the same global asymptotic stability of the rarefaction wave to the generalized Rosenau–Benjamin–Bona–Mahony–Burgers equation with a third-order dispersive term as the former one.

We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection $\Gamma$, with components $\Gamma \in L^{2p}$ and components of its Riemann curvature $\operatorname{Riem}(\Gamma)$ in $L^p$, in some coordinate system, can be smoothed by coordinate transformation to optimal regularity, $\Gamma \in W^{1,p}$ (one derivative smoother than the curvature), $p \gt \max \{ n / 2, 2 \}$, dimension $n \geq 2$. For Lorentzian metrics in General Relativity this implies that shock wave solutions of the Einstein–Euler equations are non-singular—geodesic curves, locally inertial coordinates and the Newtonian limit, all exist in a classical sense, and the Einstein equations hold in the strong sense. The proof is based on an $L^p$ existence theory for the Regularity Transformation (RT) equations, a system of elliptic partial differential equations (introduced by the authors) which determine the Jacobians of the regularizing coordinate transformations. Secondly, this existence theory gives the first extension of Uhlenbeck compactness from Riemannian metrics, to general affine connections bounded in $L^\infty$, with curvature in $L^p , p \gt n$, including semi-Riemannian metrics, and Lorentzian metric connections of relativistic Physics. We interpret this as a “geometric” improvement of the generalized Div‑Curl Lemma. Our theory shows that Uhlenbeck compactness and optimal regularity are pure logical consequences of the rule which defines how connections transform from one coordinate system to another—what one could take to be the “starting assumption of geometry”.

We prove in this note the local (in time) well-posedness of a broad class of $2 \times 2$ symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond et al. [2]. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.

We consider the problem of inverting the linear difference operator $\Delta_\Phi [v] = v \circ \Phi - v$ and obtaining bounds for the inverse operator, where $\Phi$ is a non-uniform shift on the circle. This represents the scalar version of a linearized difference operator arising in the construction of periodic solutions to the compressible Euler equations by Nash–Moser methods. We characterize the degeneracies in the linearized operators, thereby describing the complications that can arise in the application of Nash–Moser iteration to quasilinear problems. There are two cases, resonant and nonresonant, which correspond to the rationality or irrationality of the rotation number of $\Phi$, respectively. We introduce a solvability condition which characterizes the range of the difference operator, and obtain uniform bounds for the inverse operator $\Delta^{-1}_\Phi$ on this range in both cases, but our bounds are not immediately expressible in terms of standard $C^r$ or Sobolev norms. In the resonant case, the bound is in terms of the inverse width of “Arnold tongues”. In the non-resonant case the solvability condition simplifies and we translate our estimate into uniform estimates on Sobolev norms with a uniform loss of derivatives, as required for the Nash–Moser method. Our analysis is based on the introduction of the “ergodic norm”, which in addition provides an effective rate of convergence in the classical ergodic theorem.

The aim of the paper is threefold. First of all, since our main goal is to apply the discrete thermostatted kinetic framework to a socio-economic system whose space of microscopic states is discrete, we discuss the inverse problem (see the Introduction) in the discrete case. Next, we propose an example of a socio-economic system of the above kind, paying special attention to the choice of a particularly meaningful and plausible initial probability distribution on the state space. Finally, we sketch a first simple numerical simulation of the evolution of the system, just in order to show that the appropriate choice of initial conditions leads to a forecast of such evolution which seems to fit the experienced evolutions of western societies at present.

This manuscript focuses on in the transmission problem for one dimensional waves with time-varying weights on the frictional damping and time-varying delay. We prove global existence of solutions using Kato’s variable norm technique and we show the exponential stability by the energy method with the construction of a suitable Lyapunov functional.

A class of strong stability-preserving (SSP) high-order time discretization methods, first developed by Shu [18] and by Shu and Osher [19], has been demonstrated to be very effective in solving time-dependent partial differential equations (PDEs), especially hyperbolic conservation laws. In this paper, we consider an optimal second order SSP Runge–Kutta method, of which the spatial discretization is based on Sweby’s flux limiter construction [21] with minmod flux limiter and the $E$-scheme as the building block. For one-dimensional scalar convex conservation laws, we make minor modification to one of Yang’s convergence criteria [24] and then use it to show the entropy convergence of this SSP Runge–Kutta method.

In this paper we deal with the heat equation with drift in $L_{d+1}$. Basically, we prove that, if the free term is in $L_q$ with high enough $q$, then the equation is uniquely solvable in a rather unusual class of functions such that $\partial_t u , D^2 u \in L_p$ with $p \lt d + 1$ and $D_u \in L_q$.