Asymptotic Analysis

We establish a strong maximum principle for weak solutions of the mixed local and nonlocal p-Laplace equation − Δ p u + ( − Δ ) p s u = c ( x ) | u | p − 2 u in  Ω , where Ω ⊂ R N is an open set, p ∈ ( 1 , ∞ ) , s ∈ ( 0 , 1 ) and c ∈ C ( Ω ‾ ) .

We show that for any fixed accuracy and time length T, a finite number of T-time length pieces of the complete trajectories on the global attractor are capable of uniformly approximating all trajectories within the accuracy in the natural strong metric after sufficiently large time when the observed dissipative system is asymptotically compact. Moreover, we obtain the strong equicontinuity of all the complete trajectories on the global attractor. These results follow by proving the existence of a strongly compact strong trajectory attractor. The notion of a trajectory attractor was previously constructed for a family of auxiliary systems including the originally considered one without uniqueness. Recently, Cheskidov and the author developed a new framework called evolutionary system, with which a (weak) trajectory attractor can be actually defined for the original system. In this paper, the theory of trajectory attractors is further developed in the natural strong metric for our purpose. We then apply it to both the 2D and the 3D Navier–Stokes equations and a general nonautonomous reaction–diffusion system.

We study the asymptotic behaviour of the spectral gap of Schrödinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find different asymptotic behaviours of the gap. In some cases the gap behaves as the gap of the free Dirichlet Laplacian and in some cases it does not.

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

In the present article, we study the homogenization of a second-order elliptic PDE with oscillating coefficients in two different domains, namely a standard rectangular domain with very general oscillations and a circular type oscillating domain. Further, we consider the source term in L 1 and hence the solutions are interpreted as renormalized solutions. In the first domain, oscillations are in horizontal directions, while that of the second one is in the angular direction. To take into account the type of oscillations, we have used two different types of unfolding operators and have studied the asymptotic behavior of the renormalized solution of a second-order linear elliptic PDE with a source term in L 1 . In fact, we begin our study in oscillatory circular domain with oscillating coefficients and L 2 data which is also new in the literature. We also prove relevant strong convergence (corrector) results. We present the complete details in the context of circular domains, and sketch the proof in other domain.

We consider a quasilinear partial differential equation governed by the p-Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.

We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.

In this work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are established. Next, besides investigating the regularity properties, we show the continuity of mild solutions with respect to the initial functions and the order of the fractional derivative for both cases of delay separately.

Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain Ω and is Hölder continuous.

The aim of this paper is to provide uniform estimates for the eigenvalue spacing of one-dimensional semiclassical Schrödinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schrödinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain.