American Institute of Mathematical Sciences

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 \begin{document}$ \beta>0 $\end{document}. First, we study the existence and uniqueness of global in time weak solutions in the energy space. Thereafter, our main objective is to describe the long time behavior of these solutions. For this, we use some tools in the theory of dynamical systems to prove the existence of a global attractor, which is compact subset in the energy space attracting all the weak solutions when the time goes to infinity. Moreover, we derive an upper bound for the fractal dimension of the global attractor associated to these equations.

Finally, we find a range of values for the damping parameter \begin{document}$ \beta>0 $\end{document}, for which we are able to give an acute description of the internal structure of the global attractor. More precisely, we prove that in some cases the global attractor only contains the stationary (time-independent) solution of the damped Navier-Stokes-Bardina equations.

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 \begin{document}$ u^p $\end{document}, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that, for any \begin{document}$ p>1 $\end{document}, global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained for any \begin{document}$ p>1 $\end{document}, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast. We show that for any non trivial initial datum, for certain \begin{document}$ p $\end{document} depending on the Ricci curvature bound, global solutions cannot exist. On the other hand, for certain values of \begin{document}$ p $\end{document}, depending on the vector field \begin{document}$ b $\end{document}, global solutions exist, for sufficiently small initial data.

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 \begin{document}$ Z_2 $\end{document} symmetric Bogdanov-Takens bifurcation, the corresponding third-order normal form for partial functional differential equations (PFDEs) involving nonlocal interactions is derived, which is expressed concisely by original PFDEs' parameters, making it convenient to analyze effects of original parameters on dynamics and also to calculate normal form on computer. With the aid of these formulas, complex spatiotemporal patterns are theoretically predicted and numerically shown, including tri-stable nonuniform patterns with the shape of \begin{document}$ \cos \omega t\cos \frac{x}{l}- $\end{document}like or \begin{document}$ \cos \frac{x}{l}- $\end{document}like, which reflects the effects of nonlocal interactions, such as stabilizing spatiotemporal nonuniform patterns.

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 \begin{document}$ \theta $\end{document}-method as well as the backward differentiation formulas and the \begin{document}$ L1 $\end{document}-scheme. Fractional versions of these schemes, some of them completely new, are also shown, as well as their representation by means of the Grünwald–Letnikov fractional order derivative. Our results extend and improve some recent results on the subject and provide new insights on the basic nature of the weights that ensure maximal regularity.

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 \begin{document}$u_t + u u_x -\mathcal{L} u_x = 0$\end{document}, where \begin{document}$L$\end{document} is a general Fourier multiplier operator of order \begin{document}$\alpha \in [-1, 1]$\end{document}, \begin{document}$\alpha\neq 0$\end{document}, allow for small solutions to be extended beyond their ordinary existence time. The result is valid for a range of quadratic dispersive equations with inhomogenous symbols in the dispersive regime given by the parameter \begin{document}$\alpha$\end{document}.

We study properties of nonnegative functions satisfying (E)\begin{document}$ \;-{\Delta} u+u^p-M|\nabla u|^q = 0 $\end{document} in a domain of \begin{document}$ {\mathbb R}^N $\end{document} when \begin{document}$ p>1 $\end{document}, \begin{document}$ M>0 $\end{document} and \begin{document}$ 1<q<p $\end{document}. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents \begin{document}$ p $\end{document} and \begin{document}$ q $\end{document} and in particular according to the sign of \begin{document}$ q-\frac{2p}{p+1} $\end{document}, and when \begin{document}$ q = \frac{2p}{p+1} $\end{document}, also on the value of the parameter \begin{document}$ M $\end{document} which becomes a key element. The description of the different behaviours is made possible by a sharp analysis of the radial solutions of (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 \begin{document}$ p $\end{document}-Laplace evolution problem with multiplicative noise on a bounded domain \begin{document}$ D\subset\mathbb{R}^d $\end{document} with homogeneous Dirichlet boundary conditions for \begin{document}$ 1<p<\infty $\end{document}. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic \begin{document}$ p $\end{document}-Laplace equations with \begin{document}$ L^1 $\end{document}-initial data and study existence and uniqueness of solutions in this framework.

We show that for any ergodic Lebesgue measure preserving transformation \begin{document}$ f: [0,1) \rightarrow [0,1) $\end{document} and any decreasing sequence \begin{document}$ \{b_i\}_{i=1}^{\infty} $\end{document} of positive real numbers with divergent sum, the set

\begin{document}$ {\underset{n=1}{\overset{\infty}{\cap}} \, {\underset{i=n}{\overset{\infty}{\cup}}}\,} f^{-i}(B (R_{\alpha}^{i} x,b_i)) $\end{document}

has full Lebesgue measure for almost every \begin{document}$ x \in [0,1) $\end{document} and almost every \begin{document}$ \alpha \in [0,1) $\end{document}. Here \begin{document}$ B(x,r) $\end{document} is the ball of radius \begin{document}$ r $\end{document} centered at \begin{document}$ x \in [0,1) $\end{document} and \begin{document}$ R_{\alpha}: [0,1) \rightarrow [0,1) $\end{document} is rotation by \begin{document}$ \alpha \in [0,1) $\end{document}. As a corollary, we provide partial answer to a question asked by Chaika (Question \begin{document}$ 3 $\end{document}, [2]) in the context of interval exchange transformations.

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 \begin{document}$ |\lambda|^{2 k} $\end{document}, as \begin{document}$ k \to \infty $\end{document}, where \begin{document}$ -1 < \lambda < 1 $\end{document} is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, the scaling is instead like \begin{document}$ \frac{|\lambda|^k}{k} $\end{document}. We also show slower scaling laws are possible if the perturbation admits further degeneracies.

In this paper, we study the local behavior of positive singular solutions to the equation

\begin{document}$ \begin{equation*} (-\Delta)^{\sigma}u = u^{\frac{n}{n-2\sigma}}\quad \;{\rm{in }}\;B_{1}\backslash\{0\} \end{equation*} $\end{document}

where \begin{document}$ (-\Delta)^{\sigma} $\end{document} is the fractional Laplacian operator, \begin{document}$ 0<\sigma<1 $\end{document} and \begin{document}$ \frac{n}{n-2\sigma} $\end{document} is the critical Serrin exponent. We show that either \begin{document}$ u $\end{document} can be extended as a continuous function near the origin or there exist two positive constants \begin{document}$ c_{1} $\end{document} and \begin{document}$ c_{2} $\end{document} such that

\begin{document}$ \begin{equation*} c_{1}|x|^{2\sigma-n}(-\ln{|x|})^{-\frac{n-2\sigma}{2\sigma}}\leq u(x)\leq c_{2}|x|^{2\sigma-n}(-\ln{|x|})^{-\frac{n-2\sigma}{2\sigma}}\quad\;{\rm{in }}\; B_{1}\backslash\{0\}. \end{equation*} $\end{document}

Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [2]) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under \begin{document}$ f $\end{document} at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincaré's geometric theorem, our result also has some applications to reversible systems.

In this paper, we are concerned with the following elliptic equation

\begin{document}$ \begin{equation*} \begin{cases} -\Delta u = Q(x)u^{2^*-1 }+\varepsilon u^{s}, \; &{\;{\rm{in}}\;\; \Omega},\\ \ u>0, \; &{\;{\rm{in}}\;\; \Omega}, \\ \ u = 0, &{\;{\rm{on}}\;\; \partial \Omega}, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ N\geq 3 $\end{document}, \begin{document}$ s\in [1, 2^*-1) $\end{document} with \begin{document}$ 2^* = \frac{2N}{N-2} $\end{document}, \begin{document}$ \varepsilon>0 $\end{document}, \begin{document}$ \Omega $\end{document} is a smooth bounded domain in \begin{document}$ \mathbb{R}^N $\end{document}. Under some conditions on \begin{document}$ Q(x) $\end{document}, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997,461–483) proved that there exists a single-peak solution for small \begin{document}$ \varepsilon $\end{document} if \begin{document}$ N\geq 4 $\end{document} and \begin{document}$ s\in (1, 2^*-1) $\end{document}. And they proposed in Remark 1.7 of their paper that

\begin{document}$ \begin{array}{c} ''it\; is\; interesting\; to\; know\; the \;existence\; of\; single-peak \;solutions \; for \;small \;\varepsilon\;\\ and \;s = 1''.\end{array} $\end{document}

Also it was addressed in Remark 1.8 of their paper that

\begin{document}$ \begin{array}{c} ''the\; question\; of \;solutions \;concentrated \;at \;several\; points \;at \;the \;same \;time \;is \\ still \;open''. \end{array}$\end{document}

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 \begin{document}$ s = 1 $\end{document} or \begin{document}$ s>1 $\end{document}.

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

\begin{document}$\begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(n)\nabla c),\quad x\in \Omega, t>0,\\ u\cdot\nabla c = \Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P = \Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} \;\;\;\;\;\;\;\;\;\;\;\;(KSF)$ \end{document}

with no-flux boundary conditions for \begin{document}$ n $\end{document} and \begin{document}$ c $\end{document} as well as no-slip boundary condition for \begin{document}$ u $\end{document} in a bounded domain \begin{document}$ \Omega \subseteq \mathbb{R}^3 $\end{document} with smooth boundary. Here the nonnegative function \begin{document}$ S\in C^2(\bar{\Omega}) $\end{document} denotes the chemotactic sensitivity which fulfills

\begin{document}$ |S(n)|\leq C_S(1 + n)^{-\alpha} \; \; \; \; \text{for all}\; \; n\geq0 $\end{document}

with some \begin{document}$ C_S > 0 $\end{document} and \begin{document}$ \alpha> 0 $\end{document}. Imposing no restriction on the size of the initial data, by seeking some new functionals and using the bootstrap arguments on the system, we establish the existence and boundedness of global classical solutions to parabolic-elliptic Keller-Segel-Stokes system under the assumption \begin{document}$ \alpha> \frac{1}{2} $\end{document}. On the basis of this, we further prove that if the chemotactic coefficient \begin{document}$ C_S $\end{document} is appropriately small, the obtained solutions are shown to approach the spatially homogeneous steady state \begin{document}$ (\bar{n}_0, \bar{n}_0, 0) $\end{document} in the large time limit, where \begin{document}$ \bar{n}_0 = \frac{1}{|\Omega|}\int_{\Omega}n_0 $\end{document}, provided that merely \begin{document}$ n_0\not \equiv0 $\end{document} on \begin{document}$ \Omega $\end{document}.