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Abstract: Abstract We generalize work done by Picken (J Math Phys 31:616–638, 1990). We give a generalization of the Duistermaat–Heckman formula for oscillatory integrals on the based loop group G by using the Wiener measure. Previously work has been done by Wendt for the generic coadjoint orbit LG/T; we carry out a similar computation for the degenerate coadjoint orbit \(LG/ G \cong \Omega G\) . PubDate: 2022-08-03

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Abstract: Abstract We establish a weak–strong uniqueness result for the isentropic compressible Euler equations, that is: As long as a sufficiently regular solution exists, all energy-admissible weak solutions with the same initial data coincide with it. The main novelty in this contribution, compared to previous literature, is that we allow for possible vacuum in the strong solution. PubDate: 2022-07-29

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Abstract: Abstract We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation \([B,\nabla \times B] = 0\) . Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution \(u: S \rightarrow {\mathbb {R}}\) to the cohomological equation \(B\vert _S(u) = \partial _n B\) on a toroidal surface S mutually invariant to B and \(\nabla \times B\) . The right hand side \(\partial _n B\) is a normal surface derivative available to vector fields tangent to S. In this situation, we show that the field B on S is either identically zero or nowhere zero with \(B\vert _S/\Vert B\Vert ^2 \vert _S\) being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality \(\Vert B\Vert ^2 \vert _S\) ). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that \(B\vert _S\) itself is rectifiable. The rectifiability and semi-rectifiability of \(B\vert _S\) is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas. PubDate: 2022-07-11

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Abstract: Abstract This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation \(u_t=u_{xx}+f(u)\) on the real line whose initial data \(u_0=u(\cdot ,0)\) have finite limits \(\theta ^\pm \) as \(x\rightarrow \pm \infty \) . We assume that f is a locally Lipschitz function on \(\mathbb {R}\) satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u(x, t) as \(t\rightarrow \infty \) . In the first two parts of this series we mainly considered the cases where either \(\theta ^-\ne \theta ^+\) ; or \(\theta ^\pm =\theta _0\) and \(f(\theta _0)\ne 0\) ; or else \(\theta ^\pm =\theta _0\) , \(f(\theta _0)=0\) , and \(\theta _0\) is a stable equilibrium of the equation \({{\dot{\xi }}}=f(\xi )\) . In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of \(u(\cdot ,t)\) as \(t\rightarrow \infty \) are steady states. The limit profiles, or accumulation points, are taken in \(L^\infty _{loc}(\mathbb {R})\) . In the present paper, we take on the case that \(\theta ^\pm =\theta _0\) , \(f(\theta _0)=0\) , and \(\theta _0\) is an unstable equilibrium of the equation \({{\dot{\xi }}}=f(\xi )\) . Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on \(u(\cdot ,t)\) is that it is nonoscillatory (has only finitely many critical points) at some \(t\ge 0\) . Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal. PubDate: 2022-07-11

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Abstract: Abstract Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the Brusselator model, the Gray-Scott model, the Oregonator model and a certain predator-prey model. It is shown that the considered systems have the both smooth and discontinuous stationary solutions, however, only discontinuous ones can be stable. PubDate: 2022-07-11

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Abstract: Abstract We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation (PDE) in two independent variables. The MOC gives rise to two mutually coupled systems of ordinary differential equations (ODEs). As a special case we consider the Monge–Ampère (MA) equation, for which we present a general method of determining the location and number of required boundary conditions. We solve the systems of ODEs using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Reformulation of the Monge–Ampère equation as an integral equation yields via its residual a proxy for the error of the numerical solution. Numerical examples demonstrate the performance and convergence of the methods. PubDate: 2022-07-11

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Abstract: Abstract In this paper, we focus on the construction of a hybrid scheme for the approximation of non-Maxwellian kinetic models with uncertainties. In the context of multiagent systems, the introduction of a kernel at the kinetic level is useful to avoid unphysical interactions. The methods here proposed, combine a direct simulation Monte Carlo (DSMC) in the phase space together with stochastic Galerkin (sG) methods in the random space. The developed schemes preserve the main physical properties of the solution together with accuracy in the random space. The consistency of the methods is tested with respect to surrogate Fokker–Planck models that can be obtained in the quasi-invariant regime of parameters. Several applications of the schemes to non-Maxwellian models of multiagent systems are reported. PubDate: 2022-07-11

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Abstract: Abstract The Cauchy problem in \(\mathbb {R}^n\) , \(n\ge 1\) , for the degenerate parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$ is considered for \(p\ge 1\) . It is shown that given any positive \(f\in C^0([0,\infty ))\) and \(g\in C^0([0,\infty ))\) satisfying $$\begin{aligned} f(t)\rightarrow + \infty \quad \text{ and } \quad g(t)\rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$ one can find positive and radially symmetric continuous initial data with the property that the initial value problem for ( \(\star \) ) admits a positive classical solution such that $$\begin{aligned} t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow \infty \qquad \text{ and } \qquad \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)} \rightarrow 0 \qquad \text{ as } t\rightarrow \infty , \end{aligned}$$ but that $$\begin{aligned} \liminf _{t\rightarrow \infty } \frac{t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{f(t)} =0 \end{aligned}$$ and $$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{\Vert u(\cdot ,t)\Vert _{L^\infty (\mathbb {R}^n)}}{g(t)} =\infty . \end{aligned}$$ PubDate: 2022-07-06

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Abstract: Abstract We consider odd solutions to the Schrödinger equation with the \(L^{2}\) -supercritical power type nonlinearity in one dimensional Euclidean space. It is known that the odd solution scatters or blows up if its action is less than twice that of the ground state. In the present paper, we show that odd solutions with action twice that of the ground state scatter or blow up. PubDate: 2022-07-01

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Abstract: In the past few years deep artificial neural networks (DNNs) have been successfully employed in a large number of computational problems including, e.g., language processing, image recognition, fraud detection, and computational advertisement. Recently, it has also been proposed in the scientific literature to reformulate high-dimensional partial differential equations (PDEs) as stochastic learning problems and to employ DNNs together with stochastic gradient descent methods to approximate the solutions of such high-dimensional PDEs. There are also a few mathematical convergence results in the scientific literature which show that DNNs can approximate solutions of certain PDEs without the curse of dimensionality in the sense that the number of real parameters employed to describe the DNN grows at most polynomially both in the PDE dimension \(d \in {\mathbb {N}}\) and the reciprocal of the prescribed approximation accuracy \(\varepsilon > 0\) . One key argument in most of these results is, first, to employ a Monte Carlo approximation scheme which can approximate the solution of the PDE under consideration at a fixed space-time point without the curse of dimensionality and, thereafter, to prove then that DNNs are flexible enough to mimic the behaviour of the employed approximation scheme. Having this in mind, one could aim for a general abstract result which shows under suitable assumptions that if a certain function can be approximated by any kind of (Monte Carlo) approximation scheme without the curse of dimensionality, then the function can also be approximated with DNNs without the curse of dimensionality. It is a subject of this article to make a first step towards this direction. In particular, the main result of this paper, roughly speaking, shows that if a function can be approximated by means of some suitable discrete approximation scheme without the curse of dimensionality and if there exist DNNs which satisfy certain regularity properties and which approximate this discrete approximation scheme without the curse of dimensionality, then the function itself can also be approximated with DNNs without the curse of dimensionality. Moreover, for the number of real parameters used to describe such approximating DNNs we provide an explicit upper bound for the optimal exponent of the dimension \(d \in {\mathbb {N}}\) of the function under consideration as well as an explicit lower bound for the optimal exponent of the prescribed approximation accuracy \(\varepsilon >0\) . As an application of this result we derive that solutions of suitable Kolmogorov PDEs can be approximated with DNNs without the curse of dimensionality. PubDate: 2022-06-08

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Abstract: Abstract Halide perovskites are promising materials with many significant applications in photovoltaics and optoelectronics. Their highly dispersive permittivity relation leads to a non-linear relationship between the frequency and the wavenumber. This, in turn, means the resonance of the system is described by a highly non-linear eigenvalue problem, which is mathematically challenging to understand. In this paper, we use integral methods to quantify the resonant properties of halide perovskite nano-particles. We prove that, for arbitrarily small particles, the subwavelength resonant frequencies can be expressed in terms of the eigenvalues of the Newtonian potential associated with its shape. We also characterize the hybridized subwavelength resonant frequencies of a dimer of two halide perovskite particles. Finally, we examine the specific case of spherical resonators and demonstrate that our new results are consistent with previous works. PubDate: 2022-06-03

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Abstract: Abstract We introduce a new nonlocal calculus framework which parallels (and includes as a limiting case) the differential setting. The integral operators introduced have convolution structures and converge as the horizon of interaction shrinks to zero to the classical gradient, divergence, curl, and Laplacian. Moreover, a Helmholtz-type decomposition holds on the entire \(\mathbb {R}^n\) , so general vector fields can be decomposed into (nonlocal) divergence-free and curl-free components. We also identify the kernels of the nonlocal operators and prove additional properties towards building a nonlocal framework suitable for analysis of integro-differential systems. PubDate: 2022-05-19 DOI: 10.1007/s42985-022-00178-z

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Abstract: Abstract Consider a two-dimensional region whose boundary is a non-self-intersecting closed curve, which is called an interface. Suppose the movement of the region is governed only by forces on its boundary so that the area of the region is preserved. An area-preserving curvature flow is the special case when the force is dependent on the curvature of the interface. In a homogeneous medium, Gage showed that an initially convex interface remains convex and converges to a stationary circle. However, in applications, the medium is often not homogeneous and the interface moves towards a more favorable environment. The properties of the medium are described by a signal function that is a twice continuously differentiable function defined on the plane. This paper is devoted to proving the global existence of interfaces under the assumption that the Hessian of the signal function is negative definite. PubDate: 2022-05-18 DOI: 10.1007/s42985-022-00176-1

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Abstract: Abstract A relativistic version of the Kinetic Theory for polyatomic gas is considered and a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal modes is presented. In the first part, we prove via classical limit that the truncated system of moments dictates a precise hierarchy of moments in the classical framework. In the second part, we consider the particular physical case of fifteen moments closed via maximum entropy principle in a neighborhood of equilibrium state. We prove that this symmetric hyperbolic system satisfies all the general assumptions of some theorems that guarantee the global existence of smooth solutions for initial data sufficiently small. PubDate: 2022-05-16 DOI: 10.1007/s42985-022-00173-4

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Abstract: Abstract In this work, we consider the model of \({{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}\) isometric to \({\mathbb {R}}^3{\setminus } \{0\}\) , endowed with a metric conformally equivalent to the Euclidean metric of \({\mathbb {R}}^3\) , and we define a Gauss map for surfaces in this model likewise in the Euclidean 3-space. We show as a main result that any two minimal conformal immersions in \({{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}\) with the same non-constant Gauss map differ by only two types of ambient isometries: either \(f=({{\,\mathrm{\mathrm {Id}}\,}},T)\) , where T is a translation on \({\mathbb {R}}\) , or \(f=({\mathcal {A}},T)\) , where \({\mathcal {A}}\) denotes the antipodal map on \({\mathbb {S}}^2\) . This means that any minimal immersion is determined by its conformal structure and its Gauss map, up to those isometries. PubDate: 2022-05-16 DOI: 10.1007/s42985-022-00174-3

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Abstract: Abstract In this paper, we establish the existence of weak solutions to the ellipsoidal BGK model (ES-BGK model) of the Boltzmann equation with the correct Prandtl number, which corresponds to the case when the Knudsen parameter is \(-1/2\) . PubDate: 2022-05-16 DOI: 10.1007/s42985-022-00175-2

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Abstract: Abstract We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive the limiting equation that achieves the second order convergence, and provide a numerical recipe for computing the Robin coefficients. These coefficients are obtained by solving an auxiliary half-space equation. Numerically the half-space equation is solved by a spectral method that relies on the even-odd decomposition to eliminate corner-point singularity. Numerical evidences will be presented to justify the second order asymptotic convergence rate. PubDate: 2022-05-13 DOI: 10.1007/s42985-022-00172-5

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Abstract: Abstract A method of fundamental solutions (MFS) is proposed and analyzed for the ill-posed problem of finding the wave motion from given lateral Cauchy data in annular domains. A finite difference scheme, known as the Houbolt method, is applied for the time-discretisation rendering a sequence of elliptic systems corresponding to the number of time steps. The solution of the elliptic problems is sought as a linear combination of elements in what is known as a fundamental sequence with source points placed outside of the solution domain. Collocating on the boundary part where Cauchy data is given, a sequence of linear equations is obtained for finding the coefficients in the MFS approximation. Tikhonov regularization is employed to generate a stable solution to the obtained systems of linear equations. It is outlined that the elements in the fundamental sequence constitute a linearly independent and dense set on the boundary of the solution domain in the \(L_2\) -sense. Numerical results both in two and three-dimensional domains confirm the applicability of the proposed strategy for the considered lateral Cauchy problem for the wave equation both for exact and noisy data. PubDate: 2022-05-09 DOI: 10.1007/s42985-022-00177-0

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Abstract: Abstract We consider the initial-boundary value problem for the heat equation in the half space with an exponential nonlinear boundary condition. We prove the existence of global-in-time solutions under the smallness condition on the initial data in the Orlicz space \(\mathrm {exp}L^2({\mathbb {R}}^N_+)\) . Furthermore, we derive decay estimates and the asymptotic behavior for small global-in-time solutions. PubDate: 2022-05-06 DOI: 10.1007/s42985-022-00170-7