Asymptotic Analysis - Volume 112, issue 1-2

Authors: Barles, Guy | Ley, Olivier | Nguyen, Thi-Tuyen | Phan, Thanh Viet

Article Type: Research Article

Abstract: We study the large time behavior of solutions of first-order convex Hamilton–Jacobi Equations of Eikonal type u t + H ( x , D u ) = l ( x ) , set in the whole space R N × [ 0 , ∞ ) . We assume that l is bounded from below but may have arbitrary growth and therefore the solutions may also have arbitrary growth. A complete study of the structure of solutions of the ergodic problem H ( x , D v ) = l …( x ) + c is provided: contrarily to the periodic setting, the ergodic constant is not anymore unique, leading to different large time behavior for the solutions. We establish the ergodic behavior of the solutions of the Cauchy problem (i) when starting with a bounded from below initial condition and (ii) for some particular unbounded from below initial condition, two cases for which we have different ergodic constants which play a role. When the solution is not bounded from below, an example showing that the convergence may fail in general is provided. Show more

Keywords: Hamilton–Jacobi equations, asymptotic behavior, ergodic problem, unbounded solutions, viscosity solutions

DOI: 10.3233/ASY-181488

Citation: Asymptotic Analysis, vol. 112, no. 1-2, pp. 1-22, 2019

Authors: Bony, Jean-François | Popoff, Nicolas

Article Type: Research Article

Abstract: In this article, we consider the semiclassical Schrödinger operator P h = − h 2 Δ + V in R d with a confining non-negative potential V which vanishes, and study its low-lying eigenvalues λ k ( P h ) as h → 0 . First, we state a necessary and sufficient criterion upon V − 1 ( 0 ) for λ 1 ( …P h ) h − 2 to be bounded. When d = 1 and V − 1 ( 0 ) = { 0 } , we show that the size of the eigenvalues λ k ( P h ) for potentials monotonous on both sides of 0 is given by the length of an interval I h , determined by an implicit relation involving V and h . Next, we consider the case where V has a flat minimum, in the sense that it vanishes to infinite order. We provide the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on I h . Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions. Show more

Keywords: Semiclassical Schrödinger operator, eigenvalues asymptotic, degenerate potentials

DOI: 10.3233/ASY-181493

Citation: Asymptotic Analysis, vol. 112, no. 1-2, pp. 23-36, 2019

Authors: Ferreira, Lucas C.F. | Angulo-Castillo, Vladimir

Article Type: Research Article

Abstract: We are concerned with the 3D-Navier–Stokes equations with Coriolis force. Existence and uniqueness of global solutions in homogeneous Besov spaces are obtained for large speed of rotation. In the critical case of the regularity, we consider a suitable initial data class whose definition is based on the Stokes–Coriolis semigroup and Besov spaces. Moreover, we analyze the asymptotic behavior of solutions in that setting as the speed of rotation goes to infinity.

Keywords: Navier–Stokes equations, Coriolis force, global well-posedness, asymptotic behavior, Besov spaces

DOI: 10.3233/ASY-181496

Citation: Asymptotic Analysis, vol. 112, no. 1-2, pp. 37-58, 2019

Authors: Amirat, Youcef | Münch, Arnaud

Article Type: Research Article

Abstract: We perform the asymptotic analysis of the scalar advection-diffusion equation y t ε − ε y x x ε + M y x ε = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ) , M > 0 , with respect to the diffusion coefficient ε . We use the matched asymptotic expansion method which allows to describe the boundary layers of the solution. We then use the asymptotics to discuss the …controllability property of the solution for T ⩾ 1 / M . Show more

Keywords: Asymptotic analysis, boundary layers, singular controllability

DOI: 10.3233/ASY-181497

Citation: Asymptotic Analysis, vol. 112, no. 1-2, pp. 59-106, 2019

Authors: Gurevich, Pavel

Article Type: Research Article

Abstract: We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green’s functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike in the 1D case, the asymptotics contains a time independent term. The derivation of its spatial asymptotics is the technical core of the paper. Besides numerical applications, the obtained results play a crucial role in the analysis of spatio-temporal patterns for reaction-diffusion equations on lattices, in particular rattling patterns for hysteretic diffusion systems.

Keywords: Discrete heat kernel, Green’s function, lattice dynamics, asymptotics

DOI: 10.3233/ASY-181498

Citation: Asymptotic Analysis, vol. 112, no. 1-2, pp. 107-124, 2019