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1078-0947

eISSN:

1553-5231

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## Discrete & Continuous Dynamical Systems - A

August 2016 , Volume 36 , Issue 8

Special issue dedicated to Peter D. Lax on the occasion of his ninetieth birthday

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2016, 36(8): i-ii
doi: 10.3934/dcds.2016.36.8i

*+*[Abstract](1086)*+*[PDF](2472.6KB)**Abstract:**

The papers in this special issue of Discrete and Continuous Dynamical Systems are dedicated to Professor Peter D. Lax, of the Courant Institute, on the occasion of his ninetieth birthday, by some of his friends, associates, and students.

For more information please click the “Full Text” above.

2016, 36(8): 4077-4100
doi: 10.3934/dcds.2016.36.4077

*+*[Abstract](1331)*+*[PDF](759.0KB)**Abstract:**

We introduce a new method for functional representation of oscillatory integrals within any user-supplied accuracy. Our approach is based on robust methods for nonlinear approximation of functions via exponentials. The complexity of evaluation of the resulting representations of the oscillatory integrals does not depend or depends only mildly on the size of the parameter responsible for the oscillatory behavior.

2016, 36(8): 4101-4131
doi: 10.3934/dcds.2016.36.4101

*+*[Abstract](1255)*+*[PDF](536.4KB)**Abstract:**

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the coefficients and wavenumber $\lambda$ of the equation, there exists a function whose Fourier transform decays as $\exp(-\mu |\xi|)$ and which represents solutions of the differential equation with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$. In this article, we establish an improved existence theorem for nonoscillatory phase functions. Among other things, we show that solutions of second order linear ordinary differential equations can be represented with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$ using functions in the space of rapidly decaying Schwartz functions whose Fourier transforms are both exponentially decaying and compactly supported. These new observations are used in the analysis of a method for the numerical solution of second order ordinary differential equations whose running time is independent of the parameter $\lambda$. This algorithm will be reported at a later date.

2016, 36(8): 4133-4177
doi: 10.3934/dcds.2016.36.4133

*+*[Abstract](1774)*+*[PDF](2260.8KB)**Abstract:**

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

2016, 36(8): 4179-4211
doi: 10.3934/dcds.2016.36.4179

*+*[Abstract](1343)*+*[PDF](563.7KB)**Abstract:**

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the transformation from Eulerian to Lagrangian coordinates and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for the nonlinear equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the shock location or equivalently the entropy function along the shock-front downstream, all the physical variables can be expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at infinity can be uniquely determined with a uniform decay rate. To solve the free boundary problem, we employ the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the nonlinear equation, and then update the entropy function to prove that the updating map has a fixed point. Another advantage in our analysis is in the context of the full Euler equations so that the Bernoulli constant is allowed to change for different fluid trajectories.

2016, 36(8): 4213-4225
doi: 10.3934/dcds.2016.36.4213

*+*[Abstract](1155)*+*[PDF](360.7KB)**Abstract:**

We propose and solve a new problem for the unsteady transonic small disturbance equation. Data are given for the self-similar equation in a fixed, bounded region of similarity space, where on a part of the boundary the equation has degenerate type (a `sonic line') and on the remainder it is elliptic. Previous results on this problem have chosen data so that the solution is constant on the sonic line, but we set up a situation where the solution is not constant on the sonic part of the boundary. The solution we find is Lipschitz up to the boundary. Our solution sets the stage for resolution of some interesting Riemann problems for this equation and for other multidimensional conservation laws.

2016, 36(8): 4227-4246
doi: 10.3934/dcds.2016.36.4227

*+*[Abstract](1530)*+*[PDF](692.8KB)**Abstract:**

Importance sampling algorithms are discussed in detail, with an emphasis on implicit sampling, and applied to data assimilation via particle filters. Implicit sampling makes it possible to use the data to find high-probability samples at relatively low cost, making the assimilation more efficient. A new analysis of the feasibility of data assimilation is presented, showing in detail why feasibility depends on the Frobenius norm of the covariance matrix of the noise and not on the number of variables. A discussion of the convergence of particular particle filters follows. A major open problem in numerical data assimilation is the determination of appropriate priors; a progress report on recent work on this problem is given. The analysis highlights the need for a careful attention both to the data and to the physics in data assimilation problems.

2016, 36(8): 4247-4270
doi: 10.3934/dcds.2016.36.4247

*+*[Abstract](1240)*+*[PDF](888.5KB)**Abstract:**

We present an approach to designing arbitrarily high-order finite-volume spatial discretizations on locally-rectangular grids. It is based on the use of a simple class of high-order quadratures for computing the average of fluxes over faces. This approach has the advantage of being a variation on widely-used second-order methods, so that the prior experience in engineering those methods carries over in the higher-order case. Among the issues discussed are the basic design principles for uniform grids, the extension to locally-refined nest grid hierarchies, and the treatment of complex geometries using mapped grids, multiblock grids, and cut-cell representations.

2016, 36(8): 4271-4285
doi: 10.3934/dcds.2016.36.4271

*+*[Abstract](1424)*+*[PDF](433.5KB)**Abstract:**

This expository paper surveys the progress in a research program aiming at establishing the existence and long time behavior of $BV$ solutions to the Cauchy problem for hyperbolic systems of balance laws modeling relaxation phenomena.

2016, 36(8): 4287-4347
doi: 10.3934/dcds.2016.36.4287

*+*[Abstract](1562)*+*[PDF](1356.6KB)**Abstract:**

We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices $A = XX^*$, where $X$ is an $N \times n$ matrix with iid standard complex normal entries. Under the scaling $n = N + \lfloor \sqrt{ 4 c N} \rfloor$, $c > 0$ and $N \rightarrow \infty$, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices $A$ are all given by the Tracy--Widom distribution ($\beta = 2$). This scaling is motivated by the study of the solution of the equation $Ax=b$ using the conjugate gradient algorithm, in the case that $A$ and $b$ are random: For such a scaling the fluctuations of the halting time for the algorithm are empirically seen to be universal.

2016, 36(8): 4349-4366
doi: 10.3934/dcds.2016.36.4349

*+*[Abstract](1585)*+*[PDF](1139.6KB)**Abstract:**

The instability of the two-dimensional Poiseuille flow in a long channel and the subsequent transition is studied using a thermodynamic approach. The idea is to view the transition process as an initial value problem with the initial condition being Poiseuille flow plus noise, which is considered as our ensemble. Using the mean energy of the velocity fluctuation and the skin friction coefficient as the macrostate variable, we analyze the transition process triggered by the initial noises with different amplitudes. A first order transition is observed at the critical Reynolds number $Re_* \sim 5772$ in the limit of zero noise. An action function, which relates the mean energy with the noise amplitude, is defined and computed. The action function depends only on the Reynolds number, and represents the cost for the noise to trigger a transition from the laminar flow. The correlation function of the spatial structure is analyzed.

2016, 36(8): 4367-4382
doi: 10.3934/dcds.2016.36.4367

*+*[Abstract](1317)*+*[PDF](938.6KB)**Abstract:**

We give a principled approach for the selection of a boundary integral, retarded potential representation for the solution of scattering problems for the wave equation in an exterior domain.

2016, 36(8): 4383-4402
doi: 10.3934/dcds.2016.36.4383

*+*[Abstract](1618)*+*[PDF](10268.3KB)**Abstract:**

Our central result is a methodology for predicting mesh convergence for three dimensional (3D) turbulent combustion simulations, based on less expensive one dimensional (1D) and two dimensional (2D) simulations. We verify the prediction by comparison to a 3D finite rate chemistry simulation based on a reduced reaction mechanism, and we further verify it by comparison to a completely independent simulation of the same problem. We validate our simulation by comparison to experiment. Additionally, we assess grid requirements for finite rate chemistry with more detailed chemical reaction mechanism. In both cases, the test problem is an engineering scale study of a model scramjet combustor designed by Gamba et al. We find that the mesh requirements are not feasible for finite rate chemistry simulations of engineering scale problems with detailed reaction mechanism, as expected, but these criteria are less severe than the Kolmogorov scale.

2016, 36(8): 4403-4450
doi: 10.3934/dcds.2016.36.4403

*+*[Abstract](1148)*+*[PDF](2285.5KB)**Abstract:**

Existence and uniqueness theorems are proved for boundary value problems with trihedral corners and distinct boundary conditions on the faces. Part I treats strictly dissipative boundary conditions for symmetric hyperbolic systems with elliptic or hidden elliptic generators. Part II treats the Bérenger split Maxwell equations in three dimensions with possibly discontinuous absorptions. The discontinuity set of the absorptions or their derivatives has trihedral corners. Surprisingly, there is almost no loss of derivatives for the Bérenger split problem. Both problems have their origins in numerical methods with artificial boundaries.

2016, 36(8): 4451-4476
doi: 10.3934/dcds.2016.36.4451

*+*[Abstract](1489)*+*[PDF](2296.9KB)**Abstract:**

This paper addresses a multi-scale finite element method for second order linear elliptic equations with rough coefficients, which is based on the compactness of the solution operator, and does not depend on any scale-separation or periodicity assumption of the coefficient. We consider a special type of basis functions, the multi-scale basis, which are harmonic on each element and show that they have optimal approximation property for fixed local boundary conditions. To build the optimal local boundary conditions, we introduce a set of interpolation basis functions, and reduce our problem to approximating the interpolation residual of the solution space on each edge of the coarse mesh. And this is achieved through the singular value decompositions of some local oversampling operators. Rigorous error control can be obtained through thresholding in constructing the basis functions. The optimal interpolation basis functions are also identified and they can be constructed by solving some local least square problems. Numerical results for several problems with rough coefficients and high contrast inclusions are presented to demonstrate the capacity of our method in identifying and exploiting the compact structure of the local solution space to achieve computational savings.

2016, 36(8): 4477-4493
doi: 10.3934/dcds.2016.36.4477

*+*[Abstract](1334)*+*[PDF](405.3KB)**Abstract:**

It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularity in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomena, it remains to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also gives an affirmative answer to a strong version of a problem proposed by J.Nash in 1950s.

2016, 36(8): 4495-4516
doi: 10.3934/dcds.2016.36.4495

*+*[Abstract](1559)*+*[PDF](511.5KB)**Abstract:**

In this paper, we consider the Cauchy problem to the TROPICAL CLIMATE MODEL derived by Frierson--Majda--Pauluis in [15], which is a coupled system of the barotropic and baroclinic modes of the velocity and the typical midtropospheric temperature. The system considered in this paper has viscosities in the momentum equations, but no diffusivity in the temperature equation. We establish here the global well-posedness of strong solutions to this model. In proving the global existence of strong solutions, to overcome the difficulty caused by the absence of the diffusivity in the temperature equation, we introduce a new velocity $w$ (called the pseudo baroclinic velocity), which has more regularities than the original baroclinic mode of the velocity. An auxiliary function $\phi$, which looks like the effective viscous flux for the compressible Navier-Stokes equations, is also introduced to obtain the $L^\infty$ bound of the temperature. Regarding the uniqueness, we use the idea of performing suitable energy estimates at level one order lower than the natural basic energy estimates for the system.

2016, 36(8): 4517-4529
doi: 10.3934/dcds.2016.36.4517

*+*[Abstract](1560)*+*[PDF](418.6KB)**Abstract:**

In this paper we study the topological properties of the level sets, $ \S_{t}(u)=\left\{x:~u(x)= t \right\}$, of solutions $u$ of second order elliptic equations with vanishing zeroth order terms. We show that the total Betti number of level sets $\S_{t}$ is a uniformly bounded function of the parameter $t$. The uniform bound can be estimated in terms of the analytic coefficients as well as the generalized degrees of the corresponding solutions. Such an estimate is also valid for the nodal sets of solutions of the same type equations with zeroth order terms. In general, it is possible to derive from our analysis an estimate for the total Betti numbers of level sets, for large measure set of $t's$, when coefficients are sufficiently smooth, and therefore a $L^{p}$ bound on Betti numbers as a function of $t$. These estimates are obtained by a quantitative Stability Lemma and a quantitative Morse Lemma.

2016, 36(8): 4531-4552
doi: 10.3934/dcds.2016.36.4531

*+*[Abstract](1327)*+*[PDF](793.0KB)**Abstract:**

This paper is dedicated to Peter Lax. We recall happily Lax's interest in the cochlea (and in all things biomedical), culminating in his magical solution of one version of the cochlea problem, as detailed herein. The cochlea is a remarkable organ (more remarkable the more we learn about it) that separates sounds into their frequency components. Two features of the cochlea are the focus of this work. One is the extreme insensitivity of the wave motion that occurs in the cochlea to the manner in which the cochlea is stimulated, so much so that even the direction of wave propagation is independent of the location of the source of the incident sound. The other is that the cochlea is an active system, a distributed amplifier that pumps energy into the cochlear wave as it propagates. Remarkably, this amplification not only boosts the signal but also improves the frequency resolution of the cochlea. The active mechanism is modeled here by a negative damping term in the equations of motion, and the whole system is stable as a result of fluid viscosity despite the negative damping.

2016, 36(8): 4553-4567
doi: 10.3934/dcds.2016.36.4553

*+*[Abstract](1267)*+*[PDF](237.1KB)**Abstract:**

In this study we consider the density of motor proteins in filament bundles with polarity graded in space. We start with a microscopic model that includes information on motor binding site positions along specific filaments and on their polarities. We assume that filament length is small compared to the characteristic length scale of the bundle polarity pattern. This leads to a separation of scales between molecular motor movement within the bundle and along single fibers which we exploit to derive a drift-diffusion equation as a first order perturbation equation. The resulting drift-diffusion model reveals that drift dominates in unidirectional bundles while diffusion dominates in isotropic bundles. In general, however, those two modes of transport are balanced according to the polarity and thickness of the filament bundle. The model makes testable predictions on the dependence of the molecular motor density on filament density and polarity.

2016, 36(8): 4569-4577
doi: 10.3934/dcds.2016.36.4569

*+*[Abstract](1815)*+*[PDF](303.9KB)**Abstract:**

M.-J. Kang and one of us [2] developed a new version of the relative entropy method, which is efficient in the study of the long-time stability of extreme shocks. When a system of conservation laws is rich, we show that this can be adapted to the case of intermediate shocks.

2016, 36(8): 4579-4598
doi: 10.3934/dcds.2016.36.4579

*+*[Abstract](2315)*+*[PDF](1954.8KB)**Abstract:**

Entropy stability plays an important role in the dynamics of nonlinear systems of hyperbolic conservation laws and related convection-diffusion equations. Here we are concerned with the corresponding question of numerical entropy stability --- we review a general framework for designing entropy stable approximations of such systems. The framework, developed in [28,29] and in an ongoing series of works [30,6,7], is based on comparing numerical viscosities to certain

*entropy-conservative*schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks.

We demonstrate this approach with a host of second-- and higher--order accurate schemes, ranging from scalar examples to the systems of shallow-water, Euler and Navier-Stokes equations. We present a family of energy conservative schemes for the shallow-water equations with a well-balanced description of their steady-states. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in Euler equations, and we conclude with the computation of entropic measure-valued solutions based on the class of so-called TeCNO schemes --- arbitrarily high-order accurate, non-oscillatory and entropy stable schemes for systems of conservation laws.

2016, 36(8): 4599-4618
doi: 10.3934/dcds.2016.36.4599

*+*[Abstract](1527)*+*[PDF](503.5KB)**Abstract:**

It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. The main concern of this manuscript is numerical methods for dissipative chaotic infinite dimensional dynamical systems that are able to capture the stationary statistical properties of the underlying dynamical systems. We first survey results on temporal and spatial approximations that enjoy the desired properties. We then present a new result on fully discretized approximations of infinite dimensional dissipative chaotic dynamical systems that are able to capture asymptotically the stationary statistical properties. The main ingredients in ensuring the convergence of the long time statistical properties of the numerical schemes are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors of the numerical approximations is pre-compact in the phase space; (2) convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval $[0,1]$ modulo an initial layer, uniformly with respect to initial data from the union of the global attractors. The two conditions are reminiscent of the Lax equivalence theorem where stability and consistency are needed for the convergence of a numerical scheme. Applications to the complex Ginzburg-Landau equation and the two-dimensional Navier-Stokes equations in a periodic box are discussed.

2018 Impact Factor: 1.143

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