First-order Evolution Equations Research Articles

Raznostnye skhemy dekompozitsii na osnove rasshchepleniya resheniya i operatora zadachi

Domain decomposition methods are used for the approximate solution of boundary value problems for partial differential equations on parallel computing systems. The specifics of nonstationary problems is most completely taken into account when using noniterative domain decomposition schemes. Regionally additive schemes are constructed on the basis of various classes of splitting schemes. A new class of domain decomposition schemes with an additive representation of the solution on a new time level is distinguished that is based on splitting the domain into subdomains based on a partition of unity. An example of the Cauchy problem for first-order evolution equations with a positive self-adjoint operator in a finite-dimensional Hilbert space is considered. Unconditionally stable two- and three-level splitting schemes are constructed for the corresponding system of equations.

Shear imposed falling liquid films on a slippery substrate with Marangoni effects: Effect of odd viscosity

We investigate the behavior of a thin fluid with disrupted time-reversal symmetry on a uniformly heated inclined surface under external shear stress using modified Navier–Stokes equations, an energy conservation equation, and incorporating a Navier slip condition. Critical conditions for instability onset are determined by a linear stability analysis within the Orr–Sommerfeld framework. We derive a first-order Benney-type evolution equation to study long-wave instabilities. We find slippery substrate, imposed shear stress along the flow direction, and Marangoni number consistently destabilize the flow, while odd viscosity and imposed counter-flow shear stabilize it. A weakly nonlinear analysis using multiple scales reveal distinct zones of instability. Marangoni number, slip length, odd viscosity, and imposed shear direction significantly impact stability and instability regions. Numerical simulations of the free surface evolution equation of a flow system under consideration provide clear evidence of the contributions of thermocapillary, slip length, odd viscosity, and imposed shear direction. Furthermore, our analysis of linear and weakly nonlinear stability, as well as our numerical simulations, exhibit remarkable consistency.

Accelerated dynamics with dry friction via time scaling and averaging of doubly nonlinear evolution equations

In a Hilbert framework, for convex differentiable optimization, we analyze the long-time behavior of inertial dynamics with dry friction. In classical approaches based on asymptotically vanishing viscous damping (in accordance with Nesterov’s method), the results are expressed in terms of rapid convergence of the values of the function to its minimum value. On the other hand, dry friction induces convergence towards an approximate minimizer, typically the system stops at x when a given threshold ‖∇f(x)‖≤r is satisfied. We obtain rapid convergence results in this direction. In our approach, we start from a doubly nonlinear first-order evolution equation involving two potentials: one is the differentiable function f to be minimized, which acts on the state of the system via its gradient, and the other is the nonsmooth potential dry friction φ(x)=r‖x‖ which acts on the velocity vector via its sub-differential. To highlight the central role played by ∇f(x), we also deal with the dual formulation of these dynamics, which have a Riemannian gradient structure. We then rely on the general acceleration method recently developed by Attouch, Boţ and Nguyen, which consists in applying the method of time scaling and then averaging to a continuous differential equation of the first order in time. We thus obtain fast convergence results for second order time-evolution systems involving dry friction, asymptotically vanishing viscous damping, and Hessian-driven damping in the implicit form.

Line operators in Chern-Simons-Matter theories and Bosonization in Three Dimensions II: Perturbative analysis and all-loop resummation

We study mesonic line operators in Chern-Simons theories with bosonic or fermionic matter in the fundamental representation. In this paper, we elaborate on the classification and properties of these operators using all loop resummation of large N perturbation theory. We show that these theories possess two conformal line operators in the fundamental representation. One is a stable renormalization group fixed point, while the other is unstable. They satisfy first-order chiral evolution equations, in which a smooth variation of the path is given by a factorized product of two mesonic line operators. The boundary operators on which the lines can end are classified by their conformal dimension and transverse spin, which we compute explicitly at finite ’t Hooft coupling. We match the operators in the bosonic and fermionic theories. Finally, we extend our findings to the mass deformed theories and discover that the duality still holds true.

Tikhonov regularization of a nonhomogeneous first-order evolution equation

We consider a nonhomogeneous first-order evolution equation governed by a maximal monotone operator $ A $ in a Hilbert space in the presence of a Tikhonov regularization term. We study the existence and strong convergence of weak solutions to such systems. With boundedness conditions on the central path or on the solution trajectory, without assuming the zero set of $ A $ to be nonempty and with suitable assumptions on the Tikhonov regularization coefficient, we prove that the weak solutions to the system converge strongly to the element of least norm in the zero set of $ A $. As a consequence, we provide sufficient conditions where the boundedness of the weak solutions to the nonhomogeneous Tikhonov system is equivalent to the zero set of $ A $ to be nonempty. Our work is motivated by [4,17,22] and extends some results by those authors.

Approximate solution of the Cauchy problem for a first-order integrodifferential equation with solution derivative memory

We consider the Cauchy problem for a first-order evolution equation with memory in a finite-dimensional Hilbert space when the integral term is related to the time derivative of the solution. The main problems of the approximate solution of such nonlocal problems are due to the necessity to work with the approximate solution for all previous time moments. We propose a transformation of the first-order integrodifferential equation to a system of local evolutionary equations. We use the approach known in the theory of Volterra integral equations with an approximation of the difference kernel by the sum of exponents. We formulate a local problem for a weakly coupled system of equations with additional ordinary differential equations. We have given estimates of the stability of the solution by initial data and the right-hand side for the solution of the corresponding Cauchy problem. The primary attention is paid to constructing and investigating the stability of two-level difference schemes, which are convenient for computational implementation. The numerical solution of a two-dimensional model problem for the evolution equation of the first order, when the Laplace operator conditions the dependence on spatial variables, is presented.

On constraint preservation and strong hyperbolicity

We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the free-evolution approach, which consists in obtaining the solutions of the entire system by solving only the evolution equations. Certainly, this is valid only when the chosen initial data satisfies the constraints and the constraints are preserved in the evolution. In this paper, we establish the sufficient conditions required for the PDEs of the system to guarantee the constraint preservation. This is achieved by considering quasi-linear first-order PDEs, assuming the sufficient condition and deriving strongly hyperbolic first-order partial differential evolution equations for the constraints. We show that, in general, these constraint evolution equations correspond to a family of equations parametrized by a set of free parameters. We also explain how these parameters fix the propagation velocities of the constraints. As application examples of this framework, we study the constraint conservation of the Maxwell electrodynamics and the wave equations in arbitrary space–times. We conclude that the constraint evolution equations are unique in the Maxwell case and a family in the wave equation case.

Line Operators in Chern-Simons–Matter Theories and Bosonization in Three Dimensions

We study Chern-Simons theories at large N with either bosonic or fermionic matter in the fundamental representation. The most fundamental operators in these theories are mesonic line operators, the simplest example being Wilson lines ending on fundamentals. We classify the conformal line operators along an arbitrary smooth path as well as the spectrum of conformal dimensions and transverse spins of their boundary operators at finite 't Hooft coupling. These line operators are shown to satisfy first-order chiral evolution equations, in which a smooth variation of the path is given by a factorized product of two line operators. We argue that this equation together with the spectrum of boundary operators are sufficient to uniquely determine the expectation values of these operators. We demonstrate this by bootstrapping the two-point function of the displacement operator on a straight line. We show that the line operators in the theory of bosons and the theory of fermions satisfy the same evolution equation and have the same spectrum of boundary operators.

On Average Losses of Low-Frequency Sound in a Two-Dimensional Shallow-Water Random Waveguide

For a low-frequency sound signal propagating in a two-dimensionally inhomogeneous shallow-water waveguide, the influence of random bathymetry (fluctuating bottom boundary) was considered based on the local-mode approach and statistical modeling using first-order evolution equations. The study was carried out in shallow sea conditions corresponding to the coastal waveguides of the Russian Arctic seas. Here, a feature was the presence of an almost homogeneous water layer with various characteristics of seabed sediments. To describe the latter, a random model of the impedance was adopted. For the conditions of a strongly penetrable bottom boundary, on average, the calculations predicted adequate weak effects of bathymetry fluctuations on the average sound intensity compared to the effect of fluctuations in the sediment parameters and volumetric random inhomogeneities of the water column. In addition, it was shown that, in terms of statistics, the roughness of the bottom boundary perturbed the average sound intensity in a shallow-water waveguide differently than volumetric fluctuations in the speed of sound. The dependence of the statistical effects (the first and second moments of the signal intensity) on the parameters of the waveguide and the frequency range was studied. As a result of numerical modeling, comparative quantitative estimates of the influence of both the random roughness of the bottom interface and fluctuations of bottom sediment parameters on the average losses of the propagating signal, not presented in the literature, were obtained.

Numerical solution of the heat conduction problem with memory

It is necessary to use more general models than the classical Fourier heat conduction law to describe small-scale thermal conduction processes. The effects of heat flow memory and heat capacity memory (internal energy) in solids are considered in first-order integrodifferential evolutionary equations with difference-type kernels. The main difficulties in applying such nonlocal in-time mathematical models are associated with the need to work with a solution throughout the entire history of the process. The paper develops an approach to transforming a nonlocal problem into a computationally simpler local problem for a system of first-order evolution equations. Such a transition is applicable for heat conduction problems with memory if the relaxation functions of the heat flux and heat capacity are represented as a sum of exponentials. The correctness of the auxiliary linear problem is ensured by the obtained estimates of the stability of the solution concerning the initial data and the right-hand side in the corresponding Hilbert spaces. The study's main result is to prove the unconditional stability of the proposed two-level scheme with weights for the evolutionary system of equations for modeling heat conduction in solid media with memory. In this case, finding an approximate solution on a new level in time is not more complicated than the classical heat equation. The numerical solution of a model one-dimensional in space heat conduction problem with memory effects is presented.

Weak Gravitation in the 4+1 Formalism

The 4+1 formalism in general relativity (GR) prescribes field equations for the spacetime metric γμνx,τ which is local in the spacetime coordinates x and evolves according to an external “worldtime” τ. This formalism extends to GR the Stueckelberg Horwitz Piron (SHP) framework, developed to address the various issues known as the problem of time as they appear in electrodynamics. SHP field theories exhibit a formal 5D symmetry on (x,τ) that is strategically broken to 4+1 representations of the Lorentz group, resulting in a manifestly covariant canonical formalism describing the τ-evolution of spacetime structures as an initial value problem. Einstein equations for γμνx,τ are found by constructing a 5D pseudo-manifold (combining 4D geometry and τ-dynamics) and performing the natural foliation under broken 5D symmetry. This paper discusses weak gravitation in the 4+1 formalism, demonstrating the natural decomposition of the field equations into first-order evolution equations for the unconstrained 4D metric, and the propagation of constraints associated with the Bianchi identity.

Existence Theorem for Noninstantaneous Impulsive Evolution Equations

In this note, the variational form of the classical Lax–Milgram theorem is used for the divulgence of variational structure of the first-order noninstantaneous impulsive linear evolution equation. The existence and uniqueness of the weak solution of the problem is obtained. In future, this constructive theory can be used for the corresponding semilinear problems.

A new approach to the evolving 4+1 spacetime metric

We recently proposed [1, 2] field equations that prescribe a metric gαβ (x, τ) that is local in the spacetime coordinates x and evolves with the external “worldtime” τ of the Stueckelberg Horwitz Piron (SHP) framework. As in SHP electrodynamics, these field equations exhibit a formal 5D symmetry (α,β = 0, 1, 2, 3, 5), that is strategically broken to 4+1 representations of the Lorentz group. The resulting canonical formalism for this metric embodies a natural foliation of a 5D pseudo-manifold (encompassing both geometry and dynamics) into the τ-parameterized 4D spacetime posed in SHP theory. In this paper, we consider the linearized equations for weak gravitation in this 4+1 formalism, leading to a more straightforward and intuitive derivation of the coupled first-order evolution equations for the metric.

Splitting methods for solution decomposition in nonstationary problems

In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes are based on one or another additive splitting of the operator into “simpler” operators that are more convenient/easier for the computer implementation and use inhomogeneous (explicitly-implicit) time approximations. In this paper, a new class of splitting schemes is proposed that is characterized by an additive representation of the solution instead of the operator corresponding to the problem (called problem operator). A specific feature of the proposed splitting is that the resulting coupled equations for individual solution components consist of the time derivatives of the solution components. The proposed approaches are motivated by various applications, including multiscale methods, domain decomposition, and so on, where spatially local problems are solved and used to compute the solution. Unconditionally stable splitting schemes are constructed for a first-order evolution equation, which is considered in a finite-dimensional Hilbert space. In our splitting algorithms, we consider the decomposition of both the main operator of the system and the operator at the time derivative. Our goal is to provide a general framework that combines temporal splitting algorithms and spatial decomposition and its analysis. Applications of the framework will be studied separately.

A coupledj-mode method for sound propagation in range-dependent waveguides

The sound propagation problems in range-dependent waveguides are a common topic in underwater acoustics. The range-dependent factors, involving volumetric and bathymetric variations, significantly influence the propagation of sound energy and information. In this paper, a coupled-mode method based on the multimodal admittance method is presented for analyzing the sound propagation and scattering problems in range-dependent waveguides. The sound field is expanded in terms of a local basis with range-dependent modal amplitudes. The local basis corresponds to the transverse modes in a waveguide with constant physical parameters and constant cross section equal to the local cross section in the range-dependent waveguide. This local basis takes the advantage that it is easier to compute than the usual local modes which are the transverse modes in a waveguide with local physical parameters and constant cross-section equal to the local cross-section, especially for waveguides with complex environments. Projection of the Helmholtz equation that governs the sound pressure onto the local basis gives the second-order coupled mode equations for the modal amplitudes of the sound pressure. The correct boundary conditions are used in the derivation, giving rising to boundary matrices, in order to guarantee the conservation of energy among modes. The second-order coupled mode equations include coupled matrices and boundary matrices, which directly describe the effect of mode coupling due to contribution from volumetric variation (range-dependent physical parameters) and bathymetric variation (range-dependent boundaries). By introducing the admittance matrix, the second-order coupled mode equations are reduced to two sets of first-order evolution equations. The Magnus integration method is used to solve the first-order evolution equations. These first-order evolution equations allow us to obtain the numerical stable solutions and avoid the numerical divergence due to the exponential growth of evanescent modes. The numerical examples are presented for the waveguides with range-dependent physical parameters or range-dependent boundaries. The agreement between the results computed with the coupled mode method and COMSOL verifies the accuracy of the coupled mode method. Although the analysis and numerical implementation in this paper are based on two-dimensional waveguides in Cartesian coordinate system, it can be generally extended to study more complex waveguides.

First-order evolution equations with dynamic boundary conditions.

In this paper, we introduce a general framework to study linear first-order evolution equations on a Banach space X with dynamic boundary conditions, that is with boundary conditions containing time derivatives. Our method is based on the existence of an abstract Dirichlet operator and yields finally to equivalent systems of two simpler independent equations. In particular, we are led to an abstract Cauchy problem governed by an abstract Dirichlet-to-Neumann operator on the boundary space ∂X. Our approach is illustrated by several examples and various generalizations are indicated. This article is part of the theme issue 'Semigroup applications everywhere'.

Explicit–Implicit Schemes for First-Order Evolution Equations

Computational practice widely employs inhomogeneous time approximations in which one part of the problem operator is taken from the lower time level and the other, from the upper one. We discuss questions of constructing explicit–implicit schemes for the approximate solution of the Cauchy problem for a first-order evolution equation based on additive two-component splitting of the main problem operator. Novel explicit–implicit schemes are proposed when splitting the operator multiplying the time derivative. The stability of the proposed explicit–implicit two- and three-level operator-difference schemes is investigated.

Solvability of initial boundary value problems for non-autonomous evolution equations

The initial boundary value problems for linear non-autonomous first-order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change with t. We study existence, uniqueness and maximal regularity of solutions in Sobolev spaces. In contrast to the previous results, we use only the continuity assumption on the operators in the main part of the equation.

Sound propagation in inhomogeneous waveguides with sound-speed profiles using the multimodal admittance method

The multimodal admittance method and its improvement are presented to deal with various aspects in underwater acoustics, mostly for the sound propagation in inhomogeneous waveguides with sound-speed profiles, arbitrary-shaped liquid-like scatterers, and range-dependent environments. In all cases, the propagation problem governed by the Helmholtz equation is transformed into initial value problems of two coupled first-order evolution equations with respect to the modal components of field quantities (sound pressure and its derivative), by projecting the Helmholtz equation on a constructed orthogonal and complete local basis. The admittance matrix, which is the modal representation of Direchlet-to-Neumann operator, is introduced to compute the first-order evolution equations with no numerical instability caused by evanescent modes. The fourth-order Magnus scheme is used for the numerical integration of differential equations in the numerical implementation. The numerical experiments of sound field in underwater inhomogeneous waveguides generated by point sources are performed. Besides, the numerical results computed by simulation software COMSOL Multiphysics are given to validate the correction of the multimodal admittance method. It is shown that the multimodal admittance method is an efficient and stable numerical method to solve the wave propagation problem in inhomogeneous underwater waveguides with sound-speed profiles, liquid-like scatterers, and range-dependent environments. The extension of the method to more complicated waveguides such as horizontally stratified waveguides is available.

Boundary feedback stabilization of a Rao-Nakra sandwich beam

In this paper, we study the boundary feedback stabilization of a three-layer Rao-Nakra sandwich beam. Since we found a proper state space so that the energy of the system is dissipative. We then convert the beam system to an abstract first-order evolution equation with operator A. Afterwards, we obtain A is the infinitesimal generator of a C0 – semigroup of contractions by using semigroup theory. The boundary damping makes the system is stable of order 1/2, which is proved by using the frequency domain characterization of polynomial stability.