Asymptotic Analysis - Volume 121, issue 2

Authors: Lin, Ching-Lung | Lin, Liren | Nakamura, Gen

Article Type: Research Article

Abstract: The Born approximation and the Born sequence are considered for hyperbolic equations when we perturb their leading parts. The Born approximation is a finite successive approximation such as the finite terms Neumann series for the solution of a hyperbolic equation in terms of the smallness of the perturbation and if the successive approximation is infinitely many times, then we have the Born series. Due to the so called regularity loss for solutions of hyperbolic equations, we need to assume that data such as the inhomogeneous term of the equation, Cauchy datum and boundary datum are C ∞ …, and also they satisfy the compatibility condition of any order in order to define the Born series. Otherwise we need to smooth each term of the Born series. The convergence of the Born series and the Born series with smoothing are very natural questions to be asked. Also giving an estimate of approximating the solution for finite terms Born series is also an important question in practice. The aims of this paper are to discuss about these questions. We would like to emphasize that we found a small improvement in the usual energy estimate for solutions of an initial value problem for a hyperbolic equation, which is very useful for our aims. Since the estimate of approximation is only giving the worst estimate for the approximation, we also provide some numerical studies on these questions which are very suggestive for further theoretical studies on the Born approximation for hyperbolic equations. Show more

Keywords: Born approximation, Born sequence, smoothing, hyperbolic equation

DOI: 10.3233/ASY-201596

Citation: Asymptotic Analysis, vol. 121, no. 2, pp. 101-123, 2021

Authors: Antontsev, S. | de Oliveira, H.B. | Khompysh, Kh.

Article Type: Research Article

Abstract: A nonlinear initial and boundary-value problem for the Kelvin–Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established in (J. Math. Anal. Appl. 473 (2) (2019 ) 1122–1154). In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem …without convection, full anisotropic problem, and the problem with isotropic relaxation. Show more

Keywords: Kelvin–Voigt equations, anisotropic diffusion, anisotropic relaxation, anisotropic damping, large time behavior, blow-up

DOI: 10.3233/ASY-201597

Citation: Asymptotic Analysis, vol. 121, no. 2, pp. 125-157, 2021

Authors: Papageorgiou, Nikolaos S. | Vetro, Calogero | Vetro, Francesca

Article Type: Research Article

Abstract: We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 ) -superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

Keywords: Unbalanced growth, asymptotically big solutions, asymptotically small solutions, superlinear reaction, C-condition

DOI: 10.3233/ASY-201598

Citation: Asymptotic Analysis, vol. 121, no. 2, pp. 159-170, 2021

Authors: Tu, Son N.T.

Article Type: Research Article

Abstract: Let u ε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O ( ε ) of u ε → u as ε → 0 + for a large class of convex Hamiltonians H ( x , y , p ) in one dimension. This class includes the Hamiltonians from classical mechanics …with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n = 1 . Show more

Keywords: Cell problems, periodic homogenization, first order Hamilton–Jacobi equations, rate of convergence, viscosity solutions

DOI: 10.3233/ASY-201599

Citation: Asymptotic Analysis, vol. 121, no. 2, pp. 171-194, 2021

Authors: Kobayashi, Takayuki | Tsuda, Kazuyuki

Article Type: Research Article

Abstract: In this research, we study the global existence of solutions to the compressible Navier–Stokes–Korteweg system around a constant state. This system describes liquid-vapor type two-phase flow with a phase transition with diffuse interface. Previous works assume that pressure is a monotone function for change of density similarly to the usual compressible Navier–Stokes system. On the other hand, due to phase transition the pressure is in fact non-monotone function, and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. We show that global L 2 …solutions are available for the critical case of small data, whose momentum is in its derivative form, and obtain parabolic type decay rate of the solutions. This is proved based on the decomposition of solutions to a low frequency part and a high frequency part. Show more

Keywords: Partial differential equations, compressible Navier–Stokes–Korteweg system, global solution, time decay rate

DOI: 10.3233/ASY-201600

Citation: Asymptotic Analysis, vol. 121, no. 2, pp. 195-217, 2021