Asymptotic Analysis - Volume 133, issue 3

Authors: Mizoguchi, Noriko | Souplet, Philippe

Article Type: Research Article

Abstract: The Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation (VHJ) from stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after appearance of singularity, which occurs through gradient blow-up (GBU) on the boundary. Whereas viscosity solution theory has been extensively applied to many PDEs, there seem to be less results on refined singular behavior of solutions. Although occurrence of two types of GBU, with or without loss of boundary condition (LBC), are known, detailed behavior after GBU has remained open except for a strongly restricted special class of one dimensional solutions. In …this paper, in general dimensions, we construct solutions which undergo GBU with LBC arbitrarily many times and then recover regularity, as well as solutions without LBC at first GBU time. In one space dimension, we obtain the complete classification of viscosity solutions at each time, which extends to radial case in higher dimensions. Furthermore we show the existence of solutions exhibiting an arbitrarily given combination of GBU types with/without LBC at multiple times, in which a new type of behavior called “bouncing” is discovered. Global weak solutions of VHJ with multiple times singularity turn out to display larger variety of behaviors than in Fujita equation. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires delicate arguments. Since we do not rely on any known special solution unlike in Fujita equation, our method is expected to apply to other equations. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. Our results imply that for certain spatial distributions of rewards, if a controlled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times. Show more

Keywords: Viscous Hamilton–Jacobi equation, gradient blow-up, loss and recovery of boundary conditions, multiple time singularities, viscosity solutions, zero-number, critical parameters, stochastic control

DOI: 10.3233/ASY-221813

Citation: Asymptotic Analysis, vol. 133, no. 3, pp. 291-353, 2023

Authors: Araruna, Fágner D. | Mercado, Alberto | de Teresa, Luz

Article Type: Research Article

Abstract: In this paper we present a null controllability result for a thermoelastic Rayleigh system. Instead of working directly with the control system, we obtain the controlled system as the modulus of elasticity in shear tends to infinity in the corresponding thermoelastic Mindlin–Timoshenko system. Our results follow the seminal book of Lagnese and Lions (Rech. Math. Appl. 6 (1988 )) where the controllability of a Kirkhhoff model is proposed as the limit of a controlled Mindlin–Timoshenko one. We use estimates for some eigenvalues of the beam model that were obtained in (SIAM J. Control Optim. 47 (2008 ) …1909–1938) and the recent paper of Komornik and Tenenbaum (Evolution Equations and Control Theory 4 (3) (2015 ) 297–314) where explicit estimates for systems with real and complex eigenvalues are proposed. Show more

Keywords: Thermoelastic Rayleigh beam, Mindlin–Timoshenko system, controllability

DOI: 10.3233/ASY-221815

Citation: Asymptotic Analysis, vol. 133, no. 3, pp. 355-374, 2023

Authors: Avrin, Joel

Article Type: Research Article

Abstract: We consider both stationary and time-dependent solutions of the 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain Ω ⊂ R 3 with inhomogeneous boundary values on ∂ Ω = Γ ; here Γ is a union of disjoint surfaces Γ 0 , Γ 1 , … , Γ l . Our starting point is Leray’s classic problem, which is to find a weak solution u ∈ H 1 ( Ω ) of …the stationary problem assuming that on the boundary u = β ∈ H 1 / 2 ( Γ ) . The general flux condition ∑ j = 0 l ∫ Γ j β · n d S = 0 must be satisfied due to compatibility considerations. Early results on this problem including the initial results in (J. Math. Pures Appl. 12 (1933 ) 1–82) assumed the more restrictive flux condition ∫ Γ j β · n d S = 0 for each j = 1 , … , l . More recent results, of which those in (An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II 1994 Springer–Verlag) and (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are particularly representative, assume only the general flux condition in exchange for size restrictions on the data. In this paper we also assume only the general flux condition throughout, and for virtually the same size restrictions on the data as in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) we obtain the existence of a weak solution that matches that found in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) when the assumptions imposed here and those assumed in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are both met; additionally we demonstrate that this solution is unique. For slightly stronger size restrictions we obtain the existence and uniqueness of solutions of both Leray’s problem and global mild solutions of the corresponding time-dependent problem, while showing that both the stationary and time-dependent solutions we construct are a bit stronger than weak solutions. The settings in which we establish our results allow us to culminate our discussion by showing that our time-dependent solutions converge to each other exponentially in time, so that in particular our stationary solutions are asymptotically stable. We also discuss additional features which allow for data of increased size on certain domains, including those which are thin in a generalized sense. Show more

Keywords: Navier Stokes equations, inhomogenious boundary data, multi-connected domains, general flux condition, mild solutions, asymptotic stability

DOI: 10.3233/ASY-221816

Citation: Asymptotic Analysis, vol. 133, no. 3, pp. 375-396, 2023

Authors: Roulley, Emeric

Article Type: Research Article

Abstract: In this paper, we prove the existence of analytic relative equilibria with holes for quasi-geostrophic shallow-water equations. More precisely, using bifurcation techniques, we establish for any m large enough the existence of two branches of m -fold doubly-connected V-states bifurcating from any annulus of arbitrary size.

Keywords: V-states, vortex patches, bifurcation

DOI: 10.3233/ASY-221817

Citation: Asymptotic Analysis, vol. 133, no. 3, pp. 397-446, 2023