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Discrete & Continuous Dynamical Systems - A

January 2021 , Volume 41 , Issue 1

Special issue celebrating 25 years of excellence

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Editorial
Manuel del Pino, Shouchuan Hu and Juncheng Wei
2021, 41(1): i-ii doi: 10.3934/dcds.2020387 +[Abstract](209) +[HTML](86) +[PDF](74.61KB)
Abstract:
Generalized solutions to models of compressible viscous fluids
Anna Abbatiello, Eduard Feireisl and Antoní Novotný
2021, 41(1): 1-28 doi: 10.3934/dcds.2020345 +[Abstract](241) +[HTML](87) +[PDF](486.53KB)
Abstract:

We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak–strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non–linear viscosity tensor and non–homogeneous boundary conditions, for which the problem of existence of global–in–time weak/strong solutions is largely open.

Strongly localized semiclassical states for nonlinear Dirac equations
Thomas Bartsch and Tian Xu
2021, 41(1): 29-60 doi: 10.3934/dcds.2020297 +[Abstract](395) +[HTML](181) +[PDF](477.38KB)
Abstract:

We study semiclassical states of the nonlinear Dirac equation

where \begin{document}$ V $\end{document} is a bounded continuous potential function and the nonlinear term \begin{document}$ f(|\psi|)\psi $\end{document} is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity \begin{document}$ f(s) = s^p $\end{document}. We develop a variational method for the strongly indefinite functional associated to the problem.

Minimal collision arcs asymptotic to central configurations
Vivina Barutello, Gian Marco Canneori and Susanna Terracini
2021, 41(1): 61-86 doi: 10.3934/dcds.2020218 +[Abstract](606) +[HTML](253) +[PDF](547.47KB)
Abstract:

We are concerned with the analysis of finite time collision trajectories for a class of singular anisotropic homogeneous potentials of degree \begin{document}$ -\alpha $\end{document}, with \begin{document}$ \alpha\in(0,2) $\end{document} and their lower order perturbations. It is well known that, under reasonable generic assumptions, the asymptotic normalized configuration converges to a central configuration. Using McGehee coordinates, the flow can be extended to the collision manifold having central configurations as stationary points, endowed with their stable and unstable manifolds. We focus on the case when the asymptotic central configuration is a global minimizer of the potential on the sphere: our main goal is to show that, in a rather general setting, the local stable manifold coincides with that of the initial data of minimal collision arcs. This characterisation may be extremely useful in building complex trajectories with a broken geodesic method. The proof takes advantage of the generalised Sundman's monotonicity formula.

Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations
Jerry L. Bona, Angel Durán and Dimitrios Mitsotakis
2021, 41(1): 87-111 doi: 10.3934/dcds.2020215 +[Abstract](861) +[HTML](348) +[PDF](573.14KB)
Abstract:

Considered here are systems of partial differential equations arising in internal wave theory. The systems are asymptotic models describing the two-way propagation of long-crested interfacial waves in the Benjamin-Ono and the Intermediate Long-Wave regimes. Of particular interest will be solitary-wave solutions of these systems. Several methods of numerically approximating these solitary waves are put forward and their performance compared. The output of these schemes is then used to better understand some of the fundamental properties of these solitary waves.

The spatial structure of the systems of equations is non-local, like that of their one-dimensional, unidirectional relatives, the Benjamin-Ono and the Intermediate Long-Wave equations. As the non-local aspect is comprised of Fourier multiplier operators, this suggests the use of spectral methods for the discretization in space. Three iterative methods are proposed and implemented for approximating traveling-wave solutions. In addition to Newton-type and Petviashvili iterations, an interesting wrinkle on the usual Petviashvili method is put forward which appears to offer advantages over the other two techniques. The performance of these methods is checked in several ways, including using the approximations they generate as initial data in time-dependent codes for obtaining solutions of the Cauchy problem.

Attention is then turned to determining speed versus amplitude relations of these families of waves and their dependence upon parameters in the models. There are also provided comparisons between the unidirectional and bidirectional solitary waves. It deserves remark that while small-amplitude solitary-wave solutions of these systems are known to exist, our results suggest the amplitude restriction in the theory is artificial.

A posteriori error estimates for self-similar solutions to the Euler equations
Alberto Bressan and Wen Shen
2021, 41(1): 113-130 doi: 10.3934/dcds.2020168 +[Abstract](788) +[HTML](355) +[PDF](630.98KB)
Abstract:

The main goal of this paper is to analyze a family of "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.

Mathematical analysis of a cloud resolving model including the ice microphysics
Yining Cao, Chuck Jia, Roger Temam and Joseph Tribbia
2021, 41(1): 131-167 doi: 10.3934/dcds.2020219 +[Abstract](677) +[HTML](288) +[PDF](787.6KB)
Abstract:

We extend our study for the warm cloud model in [13] to the analysis of a more general cloud model including the ice microphysics in [28]. The moisture variables comprise water vapor, cloud condensates (cloud water, cloud ice), and cloud precipitations (rain, snow), with respective mass ratios \begin{document}$ q_v $\end{document}, \begin{document}$ q_c $\end{document} and \begin{document}$ q_p $\end{document}. A typical assumption in [13] for the calculation of condensation rate is that the warm clouds are exactly at water saturation with no supersaturation in general. When the ice microphysics are included, the situation becomes more complicated. We have to consider both the saturation mixing ratio with respect to water (\begin{document}$ q_{vw} $\end{document}) and the saturation with respect to ice (\begin{document}$ q_{vi} $\end{document}) when the temperature \begin{document}$ T $\end{document} is below the freezing point \begin{document}$ T_w $\end{document} but above the threshold \begin{document}$ T_i $\end{document} for homogeneous ice nucleation. A remedy, acceptable from the physical and mathematical viewpoints, is to define the overall saturation mixing ratio \begin{document}$ q_{vs} $\end{document} as a convex combination of \begin{document}$ q_{vw} $\end{document} and \begin{document}$ q_{vi} $\end{document}. Under this setting, supersaturation can still be avoided and we have the constraint \begin{document}$ q_v \le q_{vs} $\end{document} with \begin{document}$ q_{vs} $\end{document} depending itself on the state. Mathematically, we are led to a system of equations and inequations involving some quasi-variational inequalities for which we prove the global existence and regularity of solutions.

Function approximation via the subsampled Poincaré inequality
Yifan Chen and Thomas Y. Hou
2021, 41(1): 169-199 doi: 10.3934/dcds.2020296 +[Abstract](369) +[HTML](255) +[PDF](591.82KB)
Abstract:

Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincaré inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincaré inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincaré inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincaré inequality is proposed to address this problem; its optimality is also discussed.

Stuart-type polar vortices on a rotating sphere
Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy and Miles H. Wheeler
2021, 41(1): 201-215 doi: 10.3934/dcds.2020263 +[Abstract](535) +[HTML](218) +[PDF](1578.97KB)
Abstract:

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they can be adapted to model inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we show that the method used to investigate Stuart vortices on a fixed sphere provides insight into the dynamics of the large-scale zonal flows on a rotating sphere that model the background flow of polar vortices. Our approach takes advantage of the fact that while a sphere is spinning around its polar axis, every point on the sphere has the same angular velocity but its tangential velocity is proportional to the distance from the polar axis of rotation, so that points move fastest at the Equator and slower as we go towards the poles, both of which remain fixed.

Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps
Mark F. Demers
2021, 41(1): 217-256 doi: 10.3934/dcds.2020217 +[Abstract](578) +[HTML](257) +[PDF](741.73KB)
Abstract:

For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential decay of correlations against Hölder continuous functions. As a consequence, we also prove a lower bound on the rate of growth of periodic orbits. The main tool used in the paper is the construction of anisotropic Banach spaces of distributions on which the relevant weighted transfer operator has a spectral gap. We then construct our measure of maximal entropy by taking a product of left and right maximal eigenvectors of this operator.

Time-fractional equations with reaction terms: Fundamental solutions and asymptotics
Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci and Gianmaria Verzini
2021, 41(1): 257-275 doi: 10.3934/dcds.2020137 +[Abstract](973) +[HTML](428) +[PDF](391.13KB)
Abstract:

We analyze the fundamental solution of a time-fractional problem, establishing existence and uniqueness in an appropriate functional space.

We also focus on the one-dimensional spatial setting in the case in which the time-fractional exponent is equal to, or larger than, \begin{document}$ \frac12 $\end{document}. In this situation, we prove that the speed of invasion of the fundamental solution is at least "almost of square root type", namely it is larger than \begin{document}$ ct^\beta $\end{document} for any given \begin{document}$ c>0 $\end{document} and \begin{document}$ \beta\in\left(0,\frac12\right) $\end{document}.

Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions
João Marcos do Ó, Bruno Ribeiro and Bernhard Ruf
2021, 41(1): 277-296 doi: 10.3934/dcds.2020138 +[Abstract](1061) +[HTML](390) +[PDF](1415.31KB)
Abstract:

In this paper we deal with the following class of Hamiltonian elliptic systems

where \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document} is a bounded domain and \begin{document}$ g $\end{document} is a nonlinearity with exponential growth condition. We derive the maximal growth conditions allowed for \begin{document}$ f $\end{document}, proving that it can be of exponential type, double-exponential type, or completely arbitrary, depending on the conditions required for \begin{document}$ g $\end{document}. Under the hypothesis of arbitrary growth conditions or else when \begin{document}$ f $\end{document} has a double exponential growth, we prove existence of nontrivial solutions for the system.

$ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization
Aihua Fan, Jörg Schmeling and Weixiao Shen
2021, 41(1): 297-327 doi: 10.3934/dcds.2020363 +[Abstract](163) +[HTML](65) +[PDF](1081.09KB)
Abstract:

Given an integer \begin{document}$ q\ge 2 $\end{document} and a real number \begin{document}$ c\in [0,1) $\end{document}, consider the generalized Thue-Morse sequence \begin{document}$ (t_n^{(q;c)})_{n\ge 0} $\end{document} defined by \begin{document}$ t_n^{(q;c)} = e^{2\pi i c s_q(n)} $\end{document}, where \begin{document}$ s_q(n) $\end{document} is the sum of digits of the \begin{document}$ q $\end{document}-expansion of \begin{document}$ n $\end{document}. We prove that the \begin{document}$ L^\infty $\end{document}-norm of the trigonometric polynomials \begin{document}$ \sigma_{N}^{(q;c)} (x) : = \sum_{n = 0}^{N-1} t_n^{(q;c)} e^{2\pi i n x} $\end{document}, behaves like \begin{document}$ N^{\gamma(q;c)} $\end{document}, where \begin{document}$ \gamma(q;c) $\end{document} is equal to the dynamical maximal value of \begin{document}$ \log_q \left|\frac{\sin q\pi (x+c)}{\sin \pi (x+c)}\right| $\end{document} relative to the dynamics \begin{document}$ x \mapsto qx \mod 1 $\end{document} and that the maximum value is attained by a \begin{document}$ q $\end{document}-Sturmian measure. Numerical values of \begin{document}$ \gamma(q;c) $\end{document} can be computed.

On $ \epsilon $-escaping trajectories in homogeneous spaces
Federico Rodriguez Hertz and Zhiren Wang
2021, 41(1): 329-357 doi: 10.3934/dcds.2020365 +[Abstract](141) +[HTML](59) +[PDF](478.63KB)
Abstract:

Let \begin{document}$ G/\Gamma $\end{document} be a finite volume homogeneous space of a semisimple Lie group \begin{document}$ G $\end{document}, and \begin{document}$ \{\exp(tD)\} $\end{document} be a one-parameter \begin{document}$ \operatorname{Ad} $\end{document}-diagonalizable subgroup inside a simple Lie subgroup \begin{document}$ G_0 $\end{document} of \begin{document}$ G $\end{document}. Denote by \begin{document}$ Z_{\epsilon,D} $\end{document} the set of points \begin{document}$ x\in G/\Gamma $\end{document} whose \begin{document}$ \{\exp(tD)\} $\end{document}-trajectory has an escape for at least an \begin{document}$ \epsilon $\end{document}-portion of mass along some subsequence. We prove that the Hausdorff codimension of \begin{document}$ Z_{\epsilon,D} $\end{document} is at least \begin{document}$ c\epsilon $\end{document}, where \begin{document}$ c $\end{document} depends only on \begin{document}$ G $\end{document}, \begin{document}$ G_0 $\end{document} and \begin{document}$ \Gamma $\end{document}.

Mean equicontinuity, complexity and applications
Jie Li, Xiangdong Ye and Tao Yu
2021, 41(1): 359-393 doi: 10.3934/dcds.2020167 +[Abstract](802) +[HTML](366) +[PDF](451.25KB)
Abstract:

We will review the recent development of the research related to mean equicontinuity, focusing on its characterizations, its relationship with discrete spectrum, topo-isomorphy, and bounded complexity. Particularly, the application of the complexity function in the mean metric to the Sarnak and the logarithmic Sarnak Möbius disjointness conjecture will be addressed.

Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space
Hirokazu Ninomiya
2021, 41(1): 395-412 doi: 10.3934/dcds.2020364 +[Abstract](155) +[HTML](65) +[PDF](623.31KB)
Abstract:

The Allen–Cahn–Nagumo equation is a reaction-diffusion equation with a bistable nonlinearity. This equation appears to be simple, however, it includes a rich behavior of solutions. The Allen–Cahn–Nagumo equation features a solution that constantly maintains a certain profile and moves with a constant speed, which is referred to as a traveling wave solution. In this paper, the entire solution of the Allen–Cahn–Nagumo equation is studied in multi-dimensional space. Here an entire solution is meant by the solution defined for all time including negative time, even though it satisfies a parabolic partial differential equation. Especially, this equation admits traveling wave solutions connecting two stable states. It is known that there is an entire solution which behaves as two traveling wave solutions coming from both sides in one dimensional space and annihilating in a finite time and that this one-dimensional entire solution is unique up to the shift. Namely, this entire solution is symmetric with respect to some point. There is a natural question whether entire solutions coming from all directions in the multi-dimensional space are radially symmetric or not. To answer this question, radially asymmetric entire solutions will be constructed by using super-sub solutions.

Entire and ancient solutions of a supercritical semilinear heat equation
Peter Poláčik and Pavol Quittner
2021, 41(1): 413-438 doi: 10.3934/dcds.2020136 +[Abstract](893) +[HTML](392) +[PDF](435.3KB)
Abstract:

We consider the semilinear heat equation \begin{document}$ u_t = \Delta u+u^p $\end{document} on \begin{document}$ {\mathbb R}^N $\end{document}. Assuming that \begin{document}$ N\ge 3 $\end{document} and \begin{document}$ p $\end{document} is greater than the Sobolev critical exponent \begin{document}$ (N+2)/(N-2) $\end{document}, we examine entire solutions (classical solutions defined for all \begin{document}$ t\in {\mathbb R} $\end{document}) and ancient solutions (classical solutions defined on \begin{document}$ (-\infty,T) $\end{document} for some \begin{document}$ T<\infty $\end{document}). We prove a new Liouville-type theorem saying that if \begin{document}$ p $\end{document} is greater than the Lepin exponent \begin{document}$ p_L: = 1+6/(N-10) $\end{document} (\begin{document}$ p_L = \infty $\end{document} if \begin{document}$ N\le 10 $\end{document}), then all positive bounded radial entire solutions are steady states. The theorem is not valid without the assumption of radial symmetry; in other ranges of supercritical \begin{document}$ p $\end{document} it is known not to be valid even in the class of radial solutions. Our other results include classification theorems for nonstationary entire solutions (when they exist) and ancient solutions, as well as some applications in the theory of blowup of solutions.

Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction
Youshan Tao and Michael Winkler
2021, 41(1): 439-454 doi: 10.3934/dcds.2020216 +[Abstract](601) +[HTML](265) +[PDF](391.58KB)
Abstract:

We consider the haptotaxis system

which arises as a simplified version of a recently proposed model for oncolytic virotherapy. When posed under no-flux boundary conditions in a smoothly bounded domain \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document}, with positive parameters \begin{document}$ D_w $\end{document}, \begin{document}$ D_z $\end{document} and \begin{document}$ \beta $\end{document}, and along with initial conditions involving suitably regular data, this system is known to admit global classical solutions.

It is shown that with respect to infinite-time blow-up, this system exhibits a critical mass phenomenon related to the quantity \begin{document}$ m_c: = \frac{1}{(\beta-1)_+} $\end{document}: In fact, it is seen that each solution fulfilling \begin{document}$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) > m_c $\end{document} must be unbounded, and this is complemented by a boundedness result which inter alia asserts that for any choice of \begin{document}$ m<m_c $\end{document} one can find a nontrivial set of solutions, particularly containing spatially heterogeneous solutions, each of which is bounded though satisfying \begin{document}$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) = m $\end{document}.

Inverse problems for nonlinear hyperbolic equations
Gunther Uhlmann and Jian Zhai
2021, 41(1): 455-469 doi: 10.3934/dcds.2020380 +[Abstract](172) +[HTML](51) +[PDF](391.29KB)
Abstract:

There has been considerable progress in recent years in solving inverse problems for nonlinear hyperbolic equations. One of the striking aspects of these developments is the use of nonlinearity to get new information, which is not possible for the corresponding linear equations. We illustrate this for several examples including Einstein equations and the equations of nonlinear elasticity among others.

A brief and personal history of stochastic partial differential equations
Lorenzo Zambotti
2021, 41(1): 471-487 doi: 10.3934/dcds.2020264 +[Abstract](589) +[HTML](245) +[PDF](364.57KB)
Abstract:

We trace the evolution of the theory of stochastic partial differential equations from the foundation to its development, until the recent solution of long-standing problems on well-posedness of the KPZ equation and the stochastic quantization in dimension three.

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