Concentration phenomena for fractional magnetic NLS equations

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

Multiplicity and Concentration of Solutions for Kirchhoff Equations with Magnetic Field

In this paper, we study the following nonlinear magnetic Kirchhoff equation: { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) , \left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/% \epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\displaystyle% \phantom{}\text{in }\mathbb{R}^{3},\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3},\mathbb{C}),\end{aligned}\right. where ϵ > 0 {\epsilon>0} , a , b > 0 {a,b>0} are constants, V : ℝ 3 → ℝ {V:\mathbb{R}^{3}\rightarrow\mathbb{R}} and A : ℝ 3 → ℝ 3 {A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}} are continuous potentials, and Δ A ⁢ u {\Delta_{A}u} is the magnetic Laplace operator. Under a local assumption on the potential V, by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

Compressible flow simulation with moving geometries using the Brinkman penalization in high-order Discontinuous Galerkin

In this work we investigate the Brinkman volume penalization technique in the context of a high-order Discontinous Galerkin method to model moving wall boundaries for compressible fluid flow simulations. High-order approximations are especially of interest as they require few degrees of freedom to represent smooth solutions accurately. This reduced memory consumption is attractive on modern computing systems where the memory bandwidth is a limiting factor. Due to their low dissipation and dispersion they are also of particular interest for aeroacoustic problems. However, a major problem for the high-order discretization is the appropriate representation of wall geometries. In this work we look at the Brinkman penalization technique, which addresses this problem and allows the representation of geometries without modifying the computational mesh. The geometry is modelled as an artificial porous medium and embedded in the equations. As the mesh is independent of the geometry with this method, it is not only well suited for high-order discretizations but also for problems where the obstacles are moving. We look into the deployment of this strategy by briefly discussing the Brinkman penalization technique and its application in our solver and investigate its behavior in fundamental one-dimensional setups, such as shock reflection at a moving wall and the formation of a shock in front of a piston. This is followed by the application to setups with two and three dimensions, illustrating the method in the presence of curved surfaces.

Numerical Simulation of Dynamic Fluid-Structure Interaction with Elastic Structure–Rigid Obstacle Contact

An implicit scheme by partitioned procedures is proposed to solve a dynamic fluid–structure interaction problem in the case when the structure displacements are limited by a rigid obstacle. For the fluid equations (Sokes or Navier–Stokes), the fictitious domain method with penalization was used. The equality of the fluid and structure velocities at the interface was obtained using the penalization technique. The surface forces at the fluid–structure interface were computed using the fluid solution in the structure domain. A quadratic optimization problem with linear inequalities constraints was solved to obtain the structure displacements. Numerical results are presented.

Concentration results for a magnetic Schrödinger-Poisson system with critical growth

This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3,\\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials, f : ℝ → ℝ is a subcritical nonlinear term and is only continuous. Under a local assumption on the potential V, we use variational methods, penalization technique and Ljusternick-Schnirelmann theory to prove multiplicity and concentration of nontrivial solutions for ϵ > 0 small.

Compressible Flow Simulation with Moving Geometries using the Brinkman Penalization in High-Order Discontinuous Galerkin

In this work we investigate the Brinkman volume penalization technique in the context of a high-order Discontinous Galerkin method to model moving wall boundaries for compressible fluid flow simulations. High-order approximations are especially of interest as they require few degrees of freedom to represent smooth solutions accurately. This reduced memory consumption is attractive on modern computing systems where the memory bandwidth is a limiting factor. Due to their low dissipation and dispersion they are also of particular interest for aeroacoustic problems. However, a major problem for the high-order discretization is the appropriate representation of wall geometries. In this work we look at the Brinkman penalization technique, which addresses this problem and allows the representation of geometries without modifying the computational mesh. The geometry is modelled as an artificial porous medium and embedded in the equations. As the mesh is independent of the geometry with this method, it is not only well suited for highorder discretizations but also for problems where the obstacles are moving.We look into the deployment of this strategy by briefly discussing the Brinkman penalization technique and its application in our solver and investigate its behavior in fundamental one-dimensional setups, such as shock reflection at a moving wall and the formation of a shock in front of a piston. This is followed by the application to setups with two and three dimensions, illustrating the method in the presence of curved surfaces.

Stock Forecasting Using an Improved Version of Adaptive Group Lasso

Stock is one of the few things in the world that influence the economy of the society, hence its worth a shot and very desirable to predict a stock price in the future. Hence in this paper we propose a stock prediction model based on linear regression model. The database of the training is based on the Goldman Sachs database of stocks found from the Google. Here we choose Lasso penalization technique cause the, this performs well with the sparsity of the network, meaning when the network has less features and more observations. Here we have proposed an improved version of lasso function and have proposed an algorithm to improve the performance of the model.

Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation

In this paper, we study the following nonlinear magnetic Schrödinger-Poisson type equation $$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&\Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}),\end{aligned}\right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials. Under a local assumption on the potential V, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration properties of nontrivial solutions for ε > 0 small. In this problem, the function f is only continuous, which allow to consider larger classes of nonlinearities in the reaction.

Multiplicity and Concentration Results for a Magnetic Schrödinger Equation With Exponential Critical Growth in ℝ2

In this paper we study the following nonlinear Schrödinger equation with magnetic field $$\begin{align*} \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \end{align*}$$where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials, and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternik–Schnirelmann theory, we prove multiplicity and concentration of solutions for $\varepsilon $ small.