Discrete and Continuous Dynamical Systems
August 2022 , Volume 42 , Issue 8
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We address the Mach limit problem for the Euler equations in an exterior domain with an analytic boundary. We first prove the existence of tangential analytic vector fields for the exterior domain with constant analyticity radii and introduce an analytic norm in which we distinguish derivatives taken from different directions. Then we prove the uniform boundedness of the solutions in the analytic space on a time interval independent of the Mach number, and Mach limit holds in the analytic norm. The results extend more generally to Gevrey initial data with convergence in a Gevrey norm.
We consider a damped Navier-Stokes-Bardina model posed on the whole three-dimensional space. These equations write down as the well-know Navier-Stokes equations with an additional nonlocal operator in the nonlinear transport term, and moreover, with an additional damping term depending on a parameter
Finally, we find a range of values for the damping parameter
We consider certain elliptical subsets of the square lattice. The recurrent representative of the identity element of the sandpile group on this graph consists predominantly of a biperiodic pattern, along with some noise. We show that as the lattice spacing tends to 0, the fraction of the area taken up by the pattern in the identity element tends to 1.
We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term
A diffusive Rosenzweig-MacArthur model involving nonlocal prey competition is studied. Via considering joint effects of prey's carrying capacity and predator's diffusion rate, the first Turing (Hopf) bifurcation curve is precisely described, which can help to determine the parameter region where coexistence equilibrium is stable. Particularly, coexistence equilibrium can lose its stability through not only codimension one Turing (Hopf) bifurcation, but also codimension two Bogdanov-Takens, Turing-Hopf and Hopf-Hopf bifurcations, even codimension three Bogdanov-Takens-Hopf bifurcation, etc., thus the concept of Turing (Hopf) instability is extended to high codimension bifurcation instability, such as Bogdanov-Takens instability. To meticulously describe spatiotemporal patterns resulting from
We study discrete time maximal regularity in Lebesgue spaces of sequences for time-stepping schemes arising from Lubich's convolution quadrature method. We show minimal properties on the quadrature weights that determines a wide class of implicit schemes. For an appropriate choice of the weights, we are able to identify the
Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we supply a rigorous analytical proof that guarantees the convergence of this algorithm to the true parameter values when the system in question is the classic three-dimensional Lorenz system. Such a result appears to be the first of its kind for dynamical parameter estimation of nonlinear systems. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We moreover probe the limitations of the algorithm by identifying dynamical regimes under which certain parameters cannot be effectively inferred having only observed certain state variables. In such cases, modifications to the algorithm are proposed that ultimately result in recovery of the parameter. Lastly, computational evidence is provided that supports the efficacy of the algorithm well beyond the hypotheses specified by the theorem, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.
We show that Whitham type equations
We study properties of nonnegative functions satisfying (E)
In this paper, we study the persistence of invariant tori in nearly integrable multiscale twist mappings with intersection property and high degeneracy in the integrable part. Such results are also presented for the mappings with distinct number of angles and actions, which affirms the existence of lower-dimensional invariant tori in such mappings. Hence we establish a Moser's theorem in multiscales.
In this article we will describe a new construction for Gibbs measures for hyperbolic attractors generalizing the original construction of Sinai, Bowen and Ruelle of SRB measures. The classical construction of the SRB measure is based on pushing forward the normalized volume on a piece of unstable manifold. By modifying the density at each step appropriately we show that the resulting measure is a prescribed Gibbs measure. This contrasts with, and complements, the construction of Climenhaga-Pesin-Zelerowicz who replace the volume on the unstable manifold by a fixed reference measure. Moreover, the simplicity of our proof, which uses only explicit properties on the growth rate of unstable manifold and entropy estimates, has the additional advantage that it applies in more general settings.
We consider a
We show that for any ergodic Lebesgue measure preserving transformation
has full Lebesgue measure for almost every
Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which single-round periodic solutions are asymptotically stable. The distance of the bifurcation values from global resonance generically scales like
In this paper, we study the local behavior of positive singular solutions to the equation
Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [
In this paper, we are concerned with the following elliptic equation
Also it was addressed in Remark 1.8 of their paper that
Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether
Correction: “Technology Foundation of Guizhou Province” is corrected to “The Science and Technology Foundation of Guizhou Province" under Fund Project.
This paper is concerned with the volume-filling effect on global solvability and stabilization in a parabolic-elliptic Keller-Segel-Stokes systems
with no-flux boundary conditions for
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