Nonlinearity

The Cauchy problem in for the Keller–Segel system is considered for n ⩾ 3. Using a basic theory of local existence and maximal extensibility of classical and spatially integrable solutions as a starting point, the study provides a result on the occurrence of finite-time blow-up within considerably large sets of radially symmetric initial data, and moreover verifies that any such explosion exclusively occurs at the spatial origin. The detection of blow-up is accomplished by analyzing a relative of the well-known Keller–Segel energy inequality, involving a modification of the corresponding energy functional which, unlike the latter, can be seen to be favourably controlled from below by the corresponding dissipation rate through a certain functional inequality along trajectories.

Unstretchable thin elastic plates such as paper can be modelled as intrinsically flat W ^2,2 isometric immersions from a domain in into . In previous work it has been shown that if such an isometric immersion minimizes the elastic energy, then it is smooth away from a singular set consisting of three different subsets. In the present paper, we show that each of these singular subsets can indeed occur and that regularity may indeed fail there.

The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.

For a wide range of values of the intensity of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak-noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, which provides a possible alternative with respect to the edge-tracking algorithm.

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincaré section or completely outside a Poincaré section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give some examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.

We study the uniform boundedness of solutions to reaction–diffusion systems possessing a Lyapunov-like function and satisfying an intermediate sum condition. This significantly generalizes the mass dissipation condition in the literature and thus allows the nonlinearities to have arbitrary polynomial growth. We show that two dimensional reaction–diffusion systems, with quadratic intermediate sum conditions, have global solutions which are bounded uniformly in time. In higher dimensions, bounded solutions are obtained under the condition that the diffusion coefficients are quasi-uniform, i.e. they are close to each other. Applications include boundedness of solutions to chemical reaction networks with diffusion.

We consider the initial value problem which determines the pathlines of a two-phase flow, i.e. v = v( t, x) is a given velocity field of the type with Ω ^±( t) denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface Σ( t) at which v can have jump discontinuities. Since flows with phase change are included, the pathlines are allowed to cross or touch the interface. Imposing a kind of transversality condition at Σ( t), which is intimately related to the mass balance in such systems, we show existence and uniqueness of absolutely continuous solutions of the above ODE in case the one-sided velocity fields v ^± are continuous in ( t, x) and locally Lipschitz continuous in x on their respective domain of definition. A main step in proving this result, also interesting in itself, is to freeze the interface movement by means of a particular coordinate transform which requires a tailor-made extension of the intrinsic velocity field underlying a -family of moving hypersurfaces.

In this paper, we deal with the Chaplain–Lolas’s model of cancer invasion with tissue remodelling

We consider this problem in a bounded domain ( N = 2, 3) with zero-flux boundary conditions. We first establish the global existence and uniform boundedness of solutions. Subsequently, we also consider the large time behaviour of solutions, and show that the global classical solution ( u, v, w) strongly converges to the semi-trivial steady state in the large time limit if δ > η; and strongly converges to if δ < η. Unfortunately, for the case δ = η, we only prove that ( v, w) → (1, 0), and it is hard to obtain the large time limit of u due to lack of uniform boundedness of . It is worth noting that the large time behaviour of solutions for the chemotaxis–haptotaxis model with tissue remodelling has never been touched before, this paper is the first attempt. At last, taking advantage of the large time behaviour of solutions, we also establish the uniform boundedness of solutions in the classical sense.

This paper considers the problem: if coexistence occurs in the long run when a third species w invades an ecosystem consisting of two species u and v on , where u, v and w compete with one another. Under the assumption that the influence of w on u and v is small and other suitable conditions, we show that the three species can coexist as a non-monotone travelling wave. Such type of non-monotone waves plays an important role in the study of three-species phenomena. However, fewer results are known for the existence of such waves in the literature. Our approach, based on the method of sub- and super-solutions and bifurcation theory, provides a new approach to construct non-monotone waves of this type. Moreover, we show that the waves we construct are stable. To the best of our knowledge, this is the first rigorous result of stability for such type of waves.

This paper is concerned with the stationary problem of an aero-taxis system with physical boundary conditions proposed by Tuval et al (2005 Proc. Natl Acad. Sci. 102 2277–82) to describe the boundary layer formation in the air–fluid interface in any dimensions. By considering a special case where fluid is free, the stationary problem is essentially reduced to a singularly perturbed nonlocal semi-linear elliptic problem. Denoting the diffusion rate of oxygen by ɛ > 0, we show that the stationary problem admits a unique classical solution of boundary-layer profile as ɛ → 0, where the boundary-layer thickness is of order ɛ. When the domain is a ball, we find a refined asymptotic boundary layer profile up to the first-order approximation of ɛ by which we find that the slope of the layer profile in the immediate vicinity of the boundary decreases with respect to (w.r.t.) the curvature while the boundary-layer thickness increases w.r.t. the curvature.

In this paper we are concerned with a class of double phase energy functionals arising in the theory of transonic flows. Their main feature is that the associated Euler equation is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. After establishing a weighted inequality for the Baouendi–Grushin operator and a related compactness property, we establish the existence of stationary waves under arbitrary perturbations of the reaction.

This paper is concerned with a nonholonomic system with parametric excitation—the Chaplygin sleigh with time-varying mass distribution. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analogue of Fermi’s acceleration) in the case where excitation is achieved by means of a rotor with variable angular momentum. The existence of trajectories for which the translational velocity of the sleigh increases indefinitely and has the asymptotics is proved. In addition, it is shown that, when viscous friction with a nondegenerate Rayleigh function is added, unbounded speed-up disappears and the trajectories of the reduced system asymptotically tend to a limit cycle.

Introducing a suitable solution concept, we show that in bounded smooth domains , , the initial boundary value problem for the chemotaxis system with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any .

We analyze -normalized solutions of nonlinear Schrödinger systems of Gross–Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary conditions. We provide sufficient conditions for the existence of orbitally stable standing waves. Such waves correspond to global minimizers of the associated energy in the -subcritical and critical cases, and to local ones in the -supercritical case. Notably, our study also includes the Sobolev-critical case.

We present methods for bounding infinite-time averages in dynamical systems governed by nonlinear PDEs. The methods rely on auxiliary functionals, which are similar to Lyapunov functionals but satisfy different inequalities. The inequalities are enforced by requiring certain expressions to be sums of squares of polynomials, and the optimal choice of auxiliary functional is posed as a semidefinite program (SDP) that can be solved computationally. To formulate these SDPs we approximate the PDE by truncated systems of ODEs and proceed in one of two ways. The first approach is to compute bounds for the ODE systems, increasing the truncation order until bounds converge numerically. The second approach incorporates the ODE systems with analytical estimates on their deviation from the PDE, thereby using finite truncations to produce bounds for the full PDE. We apply both methods to the Kuramoto–Sivashinsky equation. In particular, we compute upper bounds on the spatiotemporal average of energy by employing polynomial auxiliary functionals up to degree six. The first approach is used for most computations, but a subset of results are checked using the second approach, and the results agree to high precision. These bounds apply to all odd solutions of period , where L is varied. Sharp bounds are obtained for , and trends suggest that more expensive computations would yield sharp bounds at larger L also. The bounds are known to be sharp (to within 0.1% numerical error) because they are saturated by the simplest nonzero steady states, which apparently have the largest mean energy among all odd solutions. Prior authors have conjectured that mean energy remains for since no particular solutions with larger energy have been found. Our bounds constitute the first positive evidence for this conjecture, albeit up to finite L, and they offer some guidance for analytical proofs.

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincaré section or completely outside a Poincaré section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give some examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.

We consider the initial value problem which determines the pathlines of a two-phase flow, i.e. v = v( t, x) is a given velocity field of the type with Ω ^±( t) denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface Σ( t) at which v can have jump discontinuities. Since flows with phase change are included, the pathlines are allowed to cross or touch the interface. Imposing a kind of transversality condition at Σ( t), which is intimately related to the mass balance in such systems, we show existence and uniqueness of absolutely continuous solutions of the above ODE in case the one-sided velocity fields v ^± are continuous in ( t, x) and locally Lipschitz continuous in x on their respective domain of definition. A main step in proving this result, also interesting in itself, is to freeze the interface movement by means of a particular coordinate transform which requires a tailor-made extension of the intrinsic velocity field underlying a -family of moving hypersurfaces.

This paper extends the Madelung–Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid quantum–classical Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint quantum–classical density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

We study the Hausdorff dimension of Gibbs measures with infinite entropy with respect to maps of the interval with countably many branches. We show that under simple conditions, such measures are symbolic-exact dimensional, and provide an almost sure value for the symbolic dimension. We also show that the lower local dimension dimension is almost surely equal to zero, while the upper local dimension is almost surely equal to the symbolic dimension. In particular, we prove that a large class of Gibbs measures with infinite entropy for the Gauss map have Hausdorff dimension zero and packing dimension equal to 1/2, and so such measures are not exact dimensional.

In this paper we address the problem of well-posedness of multi-dimensional topological Euler-alignment models introduced by Roman and Tadmor (2018 arXiv:1806.01371). The main result demonstrates local existence and uniqueness of classical solutions in class ( ρ, u) ∈ H ^{m+ α} × H ^m+1 on the periodic domain , where 0 < α < 2 is the order of singularity of the topological communication kernel ϕ( x, y), and m = m( n, α) is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator which render proper a priori estimates and help stabilize viscous approximation of the system.

For a wide range of values of the intensity of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak-noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, which provides a possible alternative with respect to the edge-tracking algorithm.

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the ‘turbulent’ and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre–Green–Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe–Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension d  =  1; among them are the KdV, BBM, Camassa–Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.

In many problems of classical mechanics and theoretical physics dynamics can be described as a slow evolution of periodic or quasi-periodic processes. Adiabatic invariants are approximate first integrals for such dynamics. The existence of adiabatic invariants makes dynamics close to regular. Destruction of adiabatic invariance leads to chaotic dynamics. We present a review of some mechanisms of destruction of adiabatic invariance in slow–fast Hamiltonian systems with examples from charged particle dynamics.

We study the Boussinesq approximation for rapidly rotating stably-stratified fluids in a three dimensional infinite layer with either stress-free or periodic boundary conditions in the vertical direction. For initial conditions satisfying a certain quasi-geostrophic smallness condition, we use dispersive estimates and the large rotation limit to prove global-in-time existence of solutions. We then use self-similar variable techniques to show that the barotropic vorticity converges to an Oseen vortex, while other components decay to zero. We finally use algebraically weighted spaces to determine leading order asymptotics. In particular we show that the barotropic vorticity approaches the Oseen vortex with algebraic rate while the barotropic vertical velocity and thermal fluctuations go to zero as Gaussians whose amplitudes oscillate in opposite phase of each other while decaying with an algebraic rate.

The effect of rapid oscillations, related to large dispersion terms, on the dynamics of dissipative evolution equations is studied for the model examples of the 1D complex Ginzburg–Landau and the Kuramoto–Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter.

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincaré section or completely outside a Poincaré section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give some examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.

We study the Hausdorff dimension of Gibbs measures with infinite entropy with respect to maps of the interval with countably many branches. We show that under simple conditions, such measures are symbolic-exact dimensional, and provide an almost sure value for the symbolic dimension. We also show that the lower local dimension dimension is almost surely equal to zero, while the upper local dimension is almost surely equal to the symbolic dimension. In particular, we prove that a large class of Gibbs measures with infinite entropy for the Gauss map have Hausdorff dimension zero and packing dimension equal to 1/2, and so such measures are not exact dimensional.

This paper extends the Madelung–Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid quantum–classical Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint quantum–classical density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

The Cauchy problem in for the Keller–Segel system is considered for n ⩾ 3. Using a basic theory of local existence and maximal extensibility of classical and spatially integrable solutions as a starting point, the study provides a result on the occurrence of finite-time blow-up within considerably large sets of radially symmetric initial data, and moreover verifies that any such explosion exclusively occurs at the spatial origin. The detection of blow-up is accomplished by analyzing a relative of the well-known Keller–Segel energy inequality, involving a modification of the corresponding energy functional which, unlike the latter, can be seen to be favourably controlled from below by the corresponding dissipation rate through a certain functional inequality along trajectories.

In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier–Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates between the global existence of smooth solutions for the two dimensional Navier–Stokes equation with arbitrarily large initial data, and the global existence of smooth solutions for the Navier–Stokes equation in three dimensions with small initial data in . This result states that the closer the initial data is to being two dimensional, the larger the initial data can be in while still guaranteeing the global existence of smooth solutions. In the whole space, this set of almost two dimensional initial data is unbounded in the critical space , but is bounded in the critical Besov spaces for all 2 < p ⩽ +∞. On the torus, however, this approach does give examples of arbitrarily large initial data in the endpoint Besov space that generate global smooth solutions to the Navier–Stokes equation. In addition to these new results, we will also sharpen the constants in a number of previously known estimates for the growth of solutions to the Navier–Stokes equation and clarify the relationship between certain component reduction type regularity criteria.

We consider the initial value problem which determines the pathlines of a two-phase flow, i.e. v = v( t, x) is a given velocity field of the type with Ω ^±( t) denoting the bulk phases of the two-phase fluid system under consideration. The bulk phases are separated by a moving and deforming interface Σ( t) at which v can have jump discontinuities. Since flows with phase change are included, the pathlines are allowed to cross or touch the interface. Imposing a kind of transversality condition at Σ( t), which is intimately related to the mass balance in such systems, we show existence and uniqueness of absolutely continuous solutions of the above ODE in case the one-sided velocity fields v ^± are continuous in ( t, x) and locally Lipschitz continuous in x on their respective domain of definition. A main step in proving this result, also interesting in itself, is to freeze the interface movement by means of a particular coordinate transform which requires a tailor-made extension of the intrinsic velocity field underlying a -family of moving hypersurfaces.

In this paper we present a methodology that allows the efficient computation of the topological derivative for semilinear elliptic problems within the averaged adjoint Lagrangian framework. The generality of our approach should also allow the extension to evolutionary and other nonlinear problems. Our strategy relies on a rescaled differential quotient of the averaged adjoint state variable which we show converges weakly to a function satisfying an equation defined in the whole space. A unique feature and advantage of this framework is that we only need to work with weakly converging subsequences of the differential quotient. This allows the computation of the topological sensitivity within a simple functional analytic framework under mild assumptions.

For a wide range of values of the intensity of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak-noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, which provides a possible alternative with respect to the edge-tracking algorithm.

Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial–temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.

Unstretchable thin elastic plates such as paper can be modelled as intrinsically flat W ^2,2 isometric immersions from a domain in into . In previous work it has been shown that if such an isometric immersion minimizes the elastic energy, then it is smooth away from a singular set consisting of three different subsets. In the present paper, we show that each of these singular subsets can indeed occur and that regularity may indeed fail there.