Long-time Behavior Research Articles

Long-time behavior for a non-autonomous Klein–Gordon–Schrödinger system with Yukawa coupling

Long-time behavior for a non-autonomous Klein–Gordon–Schrödinger system with Yukawa coupling

Evolution driven by the infinity fractional Laplacian

We consider the evolution problem associated to the infinity fractional Laplacian introduced by Bjorland et al. (Adv Math 230(4–6):1859–1894, 2012) as the infinitesimal generator of a non-Brownian tug-of-war game. We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions.

Diffusion-Induced Instability of the Periodic Solutions in a Reaction-Diffusion Predator-Prey Model with Dormancy of Predators

A reaction-diffusion predator-prey model with the dormancy of predators is considered in this paper. We are concerned with the long-time behaviors of the solutions of this system. We divided our investigations into two cases: for the ODEs system, we study the existence and stability of the equilibrium solutions and derive precise conditions on system parameters so that the system can undergo Hopf bifurcations around the positive equilibrium solution. Moreover, the properties of Hopf bifurcation are studied in detail. For the reaction-diffusion system, we are able to derive conditions on the diffusion coefficients so that the spatially homogeneous Hopf bifurcating periodic solutions can undergo diffusion-triggered instability. To support our theoretical analysis, we also include several numerical results.

L_p–L_q-Theory for a Quasilinear Non-isothermal Westervelt Equation

We investigate a quasilinear system consisting of the Westervelt equation from nonlinear acoustics and Pennes bioheat equation, subject to Dirichlet or Neumann boundary conditions. The concept of maximal regularity of type L_p–L_q is applied to prove local and global well-posedness. Moreover, we show by a parameter trick that the solutions regularize instantaneously. Finally, we compute the equilibria of the system and investigate the long-time behaviour of solutions starting close to equilibria.

Generalized solution and eventual smoothness in a logarithmic Keller–Segel system for criminal activities

This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well.

Long-time behavior of subcritical 2D vorticity Boussinesq equations

In this paper the subcritical Boussinesq system with fractional dissipation in T2 is considered. Our aim is to study the long-time behavior of solutions of Boussinesq system in its natural scale-invariant Sobolev space and prove the existence of a global attractor of optimal regularity. To this end we investigate the global well-posedness and global attractor for Boussinesq system in H2−α(T2)×H2−α(T2) via commutator estimates for nonlinear terms and a new energy estimate in Sobolev spaces to bootstrap the regularity, derived by means of nonlinear lower bounds on the fractional Laplacian. This estimate allows a sharp use of the subcritical nature of the L∞ bounds for this problem. Besides, we study the critical limit of the attractors and prove their stability and upper semicontinuity when α→1+.

Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures

We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data are not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of its physical measure. Thus, we fit physical measures by taking the Wasserstein distance from optimal transportation as a misfit function between two probability distributions. In particular, we analyze the regularity of the resulting loss function for general transportation costs and derive gradient formulas. Physical measures are approximated as fixed points of suitable PDE-based Perron–Frobenius operators. Test cases discussed in the paper include common low-dimensional dynamical systems.

Blow-up for compressible Euler system with space-dependent damping in 1-D

AbstractThis article considers the Cauchy problem for compressible Euler system inR{\bf{R}}with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs toL1(R){L}^{1}\left({\bf{R}})), then the damping is non-effective to the long-time behavior of the solution.

On long-time asymptotics to the nonlocal short pulse equation with the Schwartz-type initial data: Without solitons

In this work, the initial value problem of the nonlocal short pulse (NSP) equation is studied with the Schwartz-type initial data. Our aim is to adequately study the long-time asymptotic behavior of the solution of the NSP equation in view of the initial value condition. Starting with the Lax pair of the NSP equation, we give some discussion about the NSP equation, including infinite number of conservation laws and finite-dimensional Hamiltonian function. The relevant spectral analysis is discussed, among which it is important to note that we discuss it separately due to the singularity of the spectrum. According to these results, a suitable Riemann–Hilbert (RH) problem which is used to express the solution of the NSP equation is established. By using the nonlinear steepest descent method, the original RH problem is transformed into a model RH problem, which is given by a parabolic cylinder function. The long-time asymptotic solution of the NSP equation is therefore given, especially the error term is quite different from the local situation.

Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent

Abstract We consider the nonlinear elliptic–parabolic boundary value problem involving the Dirichlet-to-Neumann operator of p-Laplace type at the critical Sobolev exponent. We first obtain the existence and asymptotic estimates of the global solution, and the sufficient conditions of finite time blowup of the solution by using the energy method. Second, we improve the regularity of solution by Moser-type iteration. Finally, we analyze the long-time asymptotic behavior of the global solution. Moreover, with the help of the concentration compactness principle, we present a precise description of the concentration phenomenon of the solution in the forward time infinity.

The monostable cooperative system with nonlocal diffusion and free boundaries

This paper concerns the monostable cooperative system with nonlocal diffusion and free boundaries, which has recently been discussed by Du and Ni [J. Differential equations 308(2021) 369-420 and arXiv:2010.01244]. We here aim at four aspects: the first is to give more accurate estimates for the longtime behaviours of the solution; the second is to discuss the limits of solution pair of a semi-wave problem; the third is to investigate the asymptotic behaviours of the corresponding Cauchy problem; the last is to study the limiting profiles of the solution as one of the expanding rates of free boundaries converges to $\infty$. Moreover, some epidemic models are given to illustrate their own rich longtime behaviours, which are quite different from those of the relevant existing works.

Initial-boundary value problems for coupled systems of time-fractional diffusion equations

This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional diffusion equations. Defining the mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior of the solution, which mostly inherit those of a single equation. Owing to the coupling effect, we also obtain the uniqueness for an inverse problem on determining all the fractional orders by the single point observation of a single component of the solution.

Simulated diffusion spreadability for characterizing the structure and transport properties of two-phase materials

Time-dependent diffusion processes between phases of heterogeneous materials are ubiquitous in a variety of contexts in the physical, chemical, and biological sciences. Examples of such materials include composites, porous materials, geological media, cellular solids, polymer blends, colloids, gels, and biological media. The recently developed diffusion spreadability, S(t), provides a direct link between time-dependent interphase diffusive transport and the microstructure of two-phase materials across length scales (Torquato, 2021); thus making S(t) a powerful dynamic means for classifying all statistically homogeneous microstructures, spanning from anti-hyperuniform to hyperuniform. It was shown that the small-, intermediate-, and long-time behaviors of S(t) are directly determined by the small-, intermediate-, and large-scale structural features of the material. Moreover, the spreadability can be applied as a physical-property based tool for microstructural characterization in the absence of or as supplement to scattering information. In this work, we develop a computationally efficient algorithm for ascertaining S(t) directly from digitized representations of material microstructures via random-walk techniques. Our algorithm yields the time-dependent local walker concentration field c(x,t), a quantity not previously examined in the context of the spreadability, enabling us to compute the entropy production rate ṡ(t) of the associated diffusion process, which is a quantity related to the rate of energy dissipation. We also derive exact analytical expressions for ṡ(t), and find that hyperuniform materials have smaller dissipation than any nonhyperuniform materials. Lastly, we use our algorithm to compute, for the first time, the more general case of the spreadability in which the phase diffusion coefficients are distinct and provide a method for extracting the effective diffusion coefficient of the two-phase material from such data. We apply our algorithm to a variety of two- and three-dimensional simulated model (non)hyperuniform microstructures to assess their large-scale structures and diffusion properties. Given previously identified connections between the spreadability and certain nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) procedures, our algorithm is also pertinent to such experimental studies. Overall, our algorithm has practical use in the discovery and design of heterogeneous materials with desirable time-dependent diffusion properties.

Global well-posedness, scattering and blow-up for the generalized Boussinesq equation in high dimensions below the ground state energy

In this paper, we study the generalized Boussinesq equation in energy space. For spatial dimensions d≥3, nonlinear exponent α+1∈[1+4/d,1+4/(d−2)), initial data (u0,u1), we clarify the longtime behavior of the solution with energy E(u0,u1) below the ground state energy E(Qd,α,0). It depends on K(u0)=‖∇u0‖L2dα/2−2‖u0‖L22−(d−2)α/2. For K(u0)>K(Qd,α), the solution blows up in finite time. For K(u0)<K(Qd,α), the solution is global and scatters to a linear solution in energy space if u0,u1 are radial functions. To show the scattering, we use the concentration-compactness argument of Kenig-Merle [15] and the Morawetz-virial type estimate obtained in [7].

Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order

We study the long-time behavior of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model, for instance, the dynamic behavior of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power law or Ellis law for the fluid viscosity. In all three cases, positive constants (i.e., positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behavior of solutions with respect to the -norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state converge to equilibrium at rate for some . Finally, in the case of an Ellis fluid, steady states are exponentially stable in .

Long time behavior of run-and-tumble particles in two dimensions

We study the long-time asymptotic behavior of the position distribution of a run-and-tumble particle (RTP) in two dimensions in the presence of translational diffusion and show that the distribution at a time t can be expressed as a perturbative series in , where γ −1 is the persistence time of the RTP. We show that the higher order corrections to the leading order Gaussian distribution generically satisfy an inhomogeneous diffusion equation where the source term depends on the previous order solutions. The explicit solution of the inhomogeneous equation requires the position moments, and we develop a recursive formalism to compute the same. We find that the subleading corrections undergo shape transitions as the translational diffusion is increased.

On Singular Vortex Patches, I: Well-posedness Issues

The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globallymm-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long asm≥3.m\geq 3.In this case, all of the angles involved solve aclosedODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show thatanyother type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angleπ2\frac {\pi }{2}for all time. Even in the case of vortex patches with corners of angleπ2\frac {\pi }{2}or with corners which are only locallymm-fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches onR2\mathbb {R}^2with interesting dynamical behavior such as cusping and spiral formation in infinite time.

Global well-posedness and exponential decay rates of the strong solutions to the two-dimensional full compressible magnetohydrodynamics equations with vacuum in some class of large initial data

In this paper, the initial-boundary value problem for the two-dimensional viscous, compressible, and heat conducting magnetohydrodynamics equations with vacuum is considered. When all the coefficients of viscosity, heat conductivity and magnetic diffusivity are constants, first, we show that for general initial data and any time T>0, the local strong solution in the sense of Sobolev norms will never blow-up provided that the density is bounded from above and the temperature belongs to space L∞(0,T;L2)∩L2(0,T;L∞). Second, based on the above blow-up criterion and the delicate analysis of the nonlinear strong coupling equations, we establish the global existence of strong solutions when the initial mass of the fluid and the initial energy of the magnetic field (i.e., ‖ρ0‖L1 and ‖H0‖L2) are suitably small, where the refined Zlotnik inequality and the decoupling of the full compressible MHD equations play an essential role in proving the uniform upper bound of the density and the higher-order a priori uniform in time estimates, respectively. It is worth mentioning that the initial velocity and temperature can be large. Moreover, we obtain exponential decay under the H1-norm of (u,θ,H) and L2-norm of (ρu˙,ρθ˙,Ht), where the long-time behaviors of the higher-order derivative of velocity are given and it is different from the known results.

Geometric Hydrodynamics in Open Problems

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.

Dynamics of Two Interacting Kinks for the $$\phi ^{6}$$ Model

We consider the nonlinear wave equation known as the $$\phi ^{6}$$ model in dimension 1+1. We describe the long-time behavior of this model’s solutions close to a sum of two kinks with energy slightly larger than twice the minimum energy of non-constant stationary solutions. We prove orbital stability of two moving kinks. We show for low energy excess $$\epsilon $$ that these solutions can be described for a long time of order $$-\ln {(\epsilon )}\epsilon ^{-\frac{1}{2}}$$ as the sum of two moving kinks such that each kink’s center is close to an explicit function which is a solution of an ordinary differential system. We give an optimal estimate in the energy norm of the remainder and we prove that this estimate is achieved during a finite instant t of order $$-\ln {(\epsilon )}\epsilon ^{-\frac{1}{2}}.$$