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Abstract: Abstract We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence. PubDate: 2023-11-10

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Abstract: Abstract We investigate the pathwise well-posedness of stochastic partial differential equations perturbed by multiplicative Neumann boundary noise, such as fractional Brownian motion for \(H\in (1/3,1/2].\) Combining functional analytic tools with the controlled rough path approach, we establish global existence of solutions and flows for such equations. For Dirichlet boundary noise we obtain similar results for smoother noise, i.e. in the Young regime. PubDate: 2023-11-06

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Abstract: Abstract We obtain an almost sharp local well–posedness result for the biharmonic equation on the quarter plane. In addition, we prove that the nonlinear part of the solution is significantly smoother than the linear part. We use a variant of the restricted norm method of Bourgain adapted to initial–boundary value problems. Our result extends the recent results in Capistrano-Filho et al. (Pacific J Math 309(1):35–70, 2020), Ozsari and Yolcu (Commun Pure Appl Anal 18(6):3285–3316, 2019) and Basakoglu (Part Differ Equ Appl 2(4):37, 2021). It is sharp in the sense that we obtain the well–posedness threshold that was obtained for the full line problem in Seong (J Math Anal Appl 504(1):125342, 2021), with the exception of the endpoint. PubDate: 2023-10-26

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Abstract: Abstract In this paper, we propose a numerical scheme for fluid models of magnetised plasmas. One important feature of the numerical scheme is that it should be able to handle the anisotropy induced by the magnetic field. In order to do so, we propose the use of the hybrid mimetic mixed (HMM) scheme for diffusion. This is combined with a hybridised variant of the Scharfetter-Gummel (SG) scheme for advection. The proposed hybrid scheme can be implemented very efficiently via static condensation. Numerical tests are then performed to show the applicability of the combined HMM-SG scheme, even for highly anisotropic magnetic fields. PubDate: 2023-10-25

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Abstract: Abstract Along the lines of the classic Hodge–De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger’s estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to the Dirac Laplacian. A general method allowing to estimate Dirac spectral lower bounds for the Dirac spectrum of a compact Riemannian manifold in terms of the Dirac eigenvalues for a cover of 0-codimensional submanifolds is developed. Two applications are provided for the Atiyah–Singer operator. First, we prove the existence on compact connected spin manifolds of Riemannian metrics of unit volume with arbitrarily large first non zero eigenvalue, which is an already known result. Second, we prove that on a degenerating sequence of oriented, hyperbolic, three spin manifolds for any choice of the spin structures the first positive non zero eigenvalue is bounded from below by a positive uniform constant, which improves an already known result. PubDate: 2023-10-11

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Abstract: Abstract We consider a deterministic optimal control problem, focusing on a finite horizon scenario. Our proposal involves employing deep neural network approximations to capture Bellman’s dynamic programming principle. This also corresponds to solving first-order Hamilton–Jacobi–Bellman (HJB) equations. Our work builds upon the research conducted by Huré et al. (SIAM J Numer Anal 59(1):525–557, 2021), which primarily focused on stochastic contexts. However, our objective is to develop a completely novel approach specifically designed to address error propagation in the absence of diffusion in the dynamics of the system. Our analysis provides precise error estimates in terms of an average norm. Furthermore, we provide several academic numerical examples that pertain to front propagation models incorporating obstacle constraints, demonstrating the effectiveness of our approach for systems with moderate dimensions (e.g., ranging from 2 to 8) and for nonsmooth value functions. PubDate: 2023-09-26

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Abstract: Abstract We study the weak formulation of the Poisson problem on closed Lipschitz manifolds. Lipschitz manifolds do not admit tangent spaces everywhere and the definition of the Laplace–Beltrami operator is more technical than on classical differentiable manifolds (see, e.g., Gesztesy in J Math Sci 172:279–346, 2011). They however arise naturally after the triangulation of a smooth surface for computer vision or simulation purposes. We derive Stokes’ and Green’s theorems as well as a Poincaré’s inequality on Lipschitz manifolds. The existence and uniqueness of weak solutions of the Poisson problem are given in this new framework for both the continuous and discrete problems. As an example of application, numerical results are given for the Poisson problem on the boundary of the unit cube. PubDate: 2023-09-21

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Abstract: Abstract We study the regularity of solutions of parabolic partial differential equations with inhomogeneous boundary conditions on polyhedral domains \(D\subset \mathbb {R}^3\) in the specific scale \(\ B^{\alpha }_{\tau ,\tau }, \ \frac{1}{\tau }=\frac{\alpha }{3}+\frac{1}{p}\ \) of Besov spaces. The regularity of the solution in this scale determines the order of approximation that can be achieved by adaptive numerical schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our results are in good agreement with the forerunner (Dahlke and Schneider in Anal Appl 17:235–291, 2019), where parabolic equations with homogeneous boundary conditions were investigated. PubDate: 2023-09-16

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Abstract: Abstract In this paper, the concentration phenomena of the semiclassical states in the following nonlinear Dirac–Klein–Gordon system are studied $$\begin{aligned} \left\{ \begin{aligned}&-i\varepsilon \sum \limits _{k=1}^3\alpha _k\partial _k u+\omega u+m\beta u+V_1(x) u=F(x,u,\varphi ),\\&-\varepsilon ^2\Delta \varphi +\left( V_2(x)+M^2\right) \varphi =G(x,u,\varphi ), \end{aligned}\right. \end{aligned}$$ where \(u:\mathbb {R}^3\rightarrow \mathbb {C}^4\) , \(\varphi :\mathbb {R}^3\rightarrow \mathbb {R}\) , with cross nonlinearities $$\begin{aligned}&F(x,u,\varphi )=\frac{s_1}{2} u ^{s_1-2} \varphi ^{s_2}u+K(x) u ^{p-2}u,\\&G(x,u,\varphi )=s_2 u ^{s_1} \varphi ^{s_2-2}\varphi +Q(x) \varphi ^{q-2}\varphi . \end{aligned}$$ Under certain conditions, we show that the systems with \(\varepsilon >0\) small have semiclassical ground states that are concentrated around sets determined by the competing potential functions. Moreover, we also obtain some properties of these ground states as \(\varepsilon \rightarrow 0\) , such as the exponentially decay estimate. PubDate: 2023-09-09

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Abstract: Abstract We provide criteria for local unsolvability of first-order differential systems induced by complex vector fields employing techniques from the theory of locally integrable structures. Following Hörmander’s approach to study locally unsolvable equations, we obtain analogous results in the differential complex associated to a locally integrable structure provided that it is not locally exact in three different scenarios: top-degree, Levi-nondegenerate structures and co-rank 1 structures. PubDate: 2023-09-01 DOI: 10.1007/s42985-023-00260-0

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Abstract: Abstract The focus of the present work is the (theoretical) approximation of a solution of the p(x)-Poisson equation. To devise an iterative solver with guaranteed convergence, we will consider a relaxation of the original problem in terms of a truncation of the nonlinearity from below and from above by using a pair of positive cut-off parameters. We will then verify that, for any such pair, a damped Kačanov scheme generates a sequence converging to a solution of the relaxed equation. Subsequently, it will be shown that the solutions of the relaxed problems converge to the solution of the original problem in the discrete setting. Finally, the discrete solutions of the unrelaxed problem converge to the continuous solution. Our work will finally be rounded up with some numerical experiments that underline the analytical findings. PubDate: 2023-08-31 DOI: 10.1007/s42985-023-00259-7

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Abstract: Abstract We prove the energy equality of MHD system in the space founded by Cheskidov et al. (Nonlinearity 21:1233–1252, 2008) and Berselli and Chiodaroli (Nonlinear Anal 192:111704, 2020). It is clarified that the energy equality is established for a larger class of the magnetic field than that of velocity field. Most of the cases, we deal with the energy equality of MHD system in bounded domain. On the other hand, if the spacial integrability exponents of the weak solution are large, it is necessary to use the Besov space, which is suitable for us to handle freely derivatives of the nonlinear convection term. Only in this case we deal with the energy equality of MHD system in the whole space. Our result covers most of previous theorems on validity of the energy equality on the Navier–Stokes equations. PubDate: 2023-08-08 DOI: 10.1007/s42985-023-00257-9

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Abstract: Abstract High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a 2d-dimensional problem equals that of a d-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations. PubDate: 2023-08-01 DOI: 10.1007/s42985-023-00250-2

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Abstract: Abstract Recent work on path-dependent partial differential equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using regression. However, a limitation of this approach is to require the selection of a basis in a function space. In this paper, we overcome this limitation by the use of deep learning methods, and we show that this setting allows for the derivation of error bounds on the approximation of conditional expectations. Numerical examples based on a two-person zero-sum game, as well as on Asian and barrier option pricing, are presented. In comparison with other deep learning approaches, our algorithm appears to be more accurate, especially in large dimensions. PubDate: 2023-07-24 DOI: 10.1007/s42985-023-00255-x

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Abstract: Abstract We study the motion of the director field of a nematic liquid crystal submitted to a magnetic field and to a laser beam. The problem takes the form of a quasilinear wave equation in a single space variable, coupled to a Schrödinger equation. We prove the existence of a global weak solution for the initial value problem using a viscous approximation and the compensated compactness method. We finish with some numerical computations to illustrate the results. PubDate: 2023-07-21 DOI: 10.1007/s42985-023-00256-w

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Abstract: Abstract We present a new generalization of the nonlinear variational wave equation. We prove existence of local, smooth solutions for this system. As a limiting case, we recover the nonlinear variational wave equation. PubDate: 2023-07-20 DOI: 10.1007/s42985-023-00252-0

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Abstract: Abstract Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates are a priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the \(H^s\) norm during the training. PubDate: 2023-07-19 DOI: 10.1007/s42985-023-00254-y

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Abstract: Abstract In this paper, we study the following class of nonlinear Choquard equation, $$\begin{aligned} -\Delta u+a(z)u=K(u)f(u)\quad \text {in}\quad \mathbb {R}^N, \end{aligned}$$ where \(\mathbb {R}^N=\mathbb {R}^L\times \mathbb {R}^M\) , \(L\ge 2\) , \(K(u)= . ^{-\gamma }*F(u)\) , \(\gamma \in (0,N)\) , a is a continuous real function and F is the primitive function of f. Under some suitable assumptions mixed on the potential a. We prove existence of a nontrivial solution for the above equation. PubDate: 2023-07-13 DOI: 10.1007/s42985-023-00253-z

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Abstract: Abstract We investigate a class of nonlocal conservation laws in several space dimensions, where the continuum average of weighted nonlocal interactions are considered over a finite horizon. We establish well-posedness for a broad class of flux functions and initial data via semigroup theory in Banach spaces and, in particular, via the celebrated Crandall–Liggett Theorem. We also show that the unique mild solution satisfies a Kružkov-type nonlocal entropy inequality. Similarly to the local case, we demonstrate an efficient way of proving various desirable qualitative properties of the unique solution. PubDate: 2023-07-08 DOI: 10.1007/s42985-023-00249-9

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Abstract: Abstract The aim of this work is to study a new coupled fourth-order reaction-diffusion system, applied to image decomposition into cartoons and textures. The existence and uniqueness of an entropy solution to the system with initial data BH are established using Galerkin’s method. Then, numerical experiments and comparisons with other models have been performed to show the efficiency of the proposed model in image decomposition. PubDate: 2023-07-07 DOI: 10.1007/s42985-023-00251-1