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SN Partial Differential Equations and Applications (20200122) 1: 139
, January 22, 2020
By
Coclite, G. M.; Coclite, M. M.
The long time behavior of a model for the regulation of growth and patterning in developing tissues by diffusing morphogens is analyzed. Such model is expressed in terms of a system of nonlinear PDEs. The key tool in the analysis is the transformation of such system to an equation with singular diffusion.
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SN Partial Differential Equations and Applications (20200316) 1: 154
, March 16, 2020
By
Fernández, David C. Del Rey ; Carpenter, Mark H.; Dalcin, Lisandro ; Zampini, Stefano ; Parsani, Matteo
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In this paper, we extend the entropy conservative/stable algorithms presented by Del Rey Fernández et al. (2019) for the compressible Euler and Navier–Stokes equations on nonconforming prefined/coarsened curvilinear grids to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, we utilize a computationally simple and efficient approach based upon using decoupled interpolation operators. The resulting scheme is entropy conservative/stable and elementwise conservative. Numerical simulations of the isentropic vortex and viscous shock propagation confirm the entropy conservation/stability and accuracy properties of the method (achieving $$\sim p+1$$ convergence), which are comparable to those of the original conforming scheme (Carpenter et al. in SIAM J Sci Comput 36(5):B835–B867, 2014; Parsani et al. in SIAM J Sci Comput 38(5):A3129–A3162, 2016). Simulations of the Taylor–Green vortex at $$\hbox {Re}=1600$$ and turbulent flow past a sphere at $$\hbox {Re}_{\infty }=2000$$ show the robustness and stability properties of the overall spatial discretization for unstructured grids. Finally, to demonstrate the entropy conservation property of a fullydiscrete explicit entropy stable algorithm with h/p refinement/coarsening, we present the time evolution of the entropy function obtained by simulating the propagation of the isentropic vortex using a relaxation Runge–Kutta scheme.
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SN Partial Differential Equations and Applications (20200130) 1: 132
, January 30, 2020
By
Kristensson, Gerhard ; Stratis, Ioannis G.; Wellander, Niklas; Yannacopoulos, Athanasios N.
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This paper deals with the exterior Calderón operator for not necessarily spherical domains. We present a new approach of finding the norm of the exterior Calderón operator for a wide class of surfaces. The basic tool in the treatment is the set of eigenfunctions and eigenvalues to the Laplace–Beltrami operator for the surface. The norm is obtained in view of an eigenvalue problem of a quadratic form containing the exterior Calderón operator. The connection of the exterior Calderón operator to the transition matrix for a perfectly conducting surface is analyzed.
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SN Partial Differential Equations and Applications (20200113) 1: 113
, January 13, 2020
By
Estrada, Ricardo
We give a characterization of harmonic functions by a mean value type property at a single point. We show that if u is real analytic in $$\Omega ,$$ $${\mathbf {a}}$$ is a fixed point of $$\Omega ,$$ and if for all homogeneous polynomials p of degree k the one dimensional function $$\begin{aligned} \varphi _{p}\left( r\right) =\int _{\mathbb {S}}u\left( \mathbf {a+} r\omega \right) p\left( \omega \right) \,\mathrm {d} \omega, \end{aligned}$$is a polynomial of degree k at the most in some interval $$0\le r<\eta _{p},$$ then u is harmonic in $$\Omega .$$ If u is smooth, and $$\eta _{p}=\eta $$ does not depend on p, then we show that u must be harmonic in the ball of center $${\mathbf {a}}$$ and radius $$\eta .$$ We also give a result that applies to distributions. Furthermore, we characterize harmonic functions by flow integrals around a single point.
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SN Partial Differential Equations and Applications (20200123) 1: 115
, January 23, 2020
By
Bez, Neal; Machihara, Shuji ; Ozawa, Tohru
A Hardy type inequality is presented with spherical derivatives in $${\mathbb {R}}^{n}$$ with $$n\ge 2$$ in the framework of equalities. This clarifies the difference between contribution by radial and spherical derivatives in the improved Hardy inequality as well as nonexistence of nontrivial extremizers without compactness arguments.
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SN Partial Differential Equations and Applications (20200406) 1: 134
, April 06, 2020
By
Hutzenthaler, Martin ; Jentzen, Arnulf; Kruse, Thomas; Nguyen, Tuan Anh
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1 Citations
Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of highdimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs it has also been proved mathematically that deep neural networks overcome the curse of dimensionality in the numerical approximation of solutions of such linear PDEs. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradientindependent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced full history recursive multilevel Picard approximations for semilinear PDEs.
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SN Partial Differential Equations and Applications (20200414) 1: 126
, April 14, 2020
By
Abdullah Sharaf, Khadijah; Chtioui, Hichem
In this paper, we study the critical fractional nonlinear PDE: $$(\Delta )^{s}u= u^\frac{n+2s}{n2s}$$, $${u>0}$$ in $$\Omega $$ and $$u=0$$ on $$\partial \Omega $$, where $$\Omega $$ is a thin annulidomain of $${\mathbb{R}}^n, n\ge 2.$$
We compute the evaluation of the difference of topology induced by the critical points at infinity between the level sets of the associated variational function. Our Theorem can be seen as a nonlocal analog of the result of Ahmedou and El Mehdi (Duke Math J 94:215–229, 1998) on the classical Yamabetype equation.
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SN Partial Differential Equations and Applications (20200313) 1: 112
, March 13, 2020
By
Sun, Dongdong; Zhang, Zhitao
We study the existence of positive solutions to the following Kirchhoff type equation with vanishing potential and general nonlinearity: $$\begin{aligned} \left\{ \begin{aligned}&(\varepsilon ^2a+\varepsilon b{\int _{\mathbb {R}^3}}{\nabla v}^{2})\Delta v+V(x)v=f(v), ~~~~x\in \mathbb {R}^3, \\&v>0,~~~v\in H^{1}(\mathbb {R}^3), \end{aligned} \right. \end{aligned}$$where $$\varepsilon >0$$ is a small parameter, $$a,b>0$$ are constants and the potential V can vanish, i.e., the zero set of V, $$\mathcal {Z}:=\{x\in \mathbb {R}^3V(x)=0\}$$ is nonempty. In our case, the method of Nehari manifold does not work any more. We first make a truncation of the nonlinearity and prove the existence of solutions for the equation with truncated nonlinearity, then by elliptic estimates, we prove that the solution of truncated equation is just the solution of our original problem for sufficiently small $$\varepsilon >0$$.
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SN Partial Differential Equations and Applications (20200217) 1: 126
, February 17, 2020
By
Kuehn, Christian ; Soresina, Cinzia
In this paper we investigate the bifurcation structure of the crossdiffusion Shigesada–Kawasaki–Teramoto model (SKT) in the triangular form and in the weak competition regime, and of a corresponding fastreaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be exploited to treat crossdiffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the linearized analysis. In 1D and 2D, we pay particular attention to the fastreaction limit by always computing sequences of bifurcation diagrams as the timescale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fastreaction systems limits onto the crossdiffusion singular limit. Furthermore, we find evidence for timeperiodic solutions by detecting Hopf bifurcations, we characterize several regions of multistability, and improve our understanding of the shape of patterns in 2D for the SKT model.
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SN Partial Differential Equations and Applications (20200113) 1: 115
, January 13, 2020
By
Motreanu, Dumitru ; Vetro, Calogero; Vetro, Francesca
Our objective is to study a new type of Dirichlet boundary value problem consisting of a system of equations with parameters, where the reaction terms depend on both the solution and its gradient (i.e., they are convection terms) and incorporate the effects of convolutions. We present results on existence, uniqueness and dependence of solutions with respect to the parameters involving convolutions.
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