Related PDE Research Articles

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

Nonlocal symmetries, nonlocally related systems, similarity solutions and conservation laws of the Tzitzéica-Dodd-Bullogh equation

In this paper, a methodical procedure is utilized for the identification of nonlocal symmetries of the (1+1)-dimensional Tzitzéica-Dodd-Bullogh equation. Firstly, by introducing a set of canonical coordinates corresponding to the local Lie point symmetries, the considered partial differential eq. (PDE) is mapped to an invertibly equivalent PDE system. Furthermore, nonlocal symmetries are obtained from the inverse potential system of the invertibly equivalent PDE system. The exact solutions for the aforementioned PDE are acquired with the help of the extended generalized Kudryashov method corresponding to the admitted symmetries. In addition, the derivation of local conservation laws for the Tzitzéica-Dodd-Bullogh equation is obtained through the multiplier method. Moreover, using a symmetry-based technique and local conservation principles, a complete tree of nonlocally related PDE systems has been constructed.

A paradifferential approach for hyperbolic dynamical systems and applications

We develop a paradifferential approach for studying non-smooth hyperbolic dynamics and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the $H^s$ wave-front set for all $s$, of the unstable bundle $E_u$ for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than H\"older: there is $s_0>0$ such that if $E_u$ has $H^s$ regularity for $s>s_0$ then it is smooth (with $s_0=2$ for volume preserving $3$-dimensional Anosov flows). In the appendix by Guedes Bonthonneau, it is also shown that it can be applied to deal with non-smooth flows and potentials. This work could serve as a toolbox for other applications.

Global existence for a highly nonlinear temperature-dependent system modeling nonlocal adhesive contact

In this paper we analyze a new temperature-dependent model for adhesive contact that encompasses nonlocal adhesive forces and damage effects, as well as nonlocal heat flux contributions on the contact surface. The related PDE system combines heat equations, in the bulk domain and on the contact surface, with mechanical force balances, including micro-forces, that result in the equation for the displacements and in the flow rule for the damage-type internal variable describing the state of the adhesive bonds. Nonlocal effects are accounted for by terms featuring integral operators on the contact surface.The analysis of this system poses several difficulties due to its overall highly nonlinear character, and in particular to the presence of quadratic terms, in the rates of the strain tensor and of the internal variable, that appear in the bulk and surface heat equations. Another major challenge is related to proving strict positivity for the bulk and surface temperatures.We tackle these issues by very careful estimates that enable us to prove the existence of global-in-time solutions and could be useful in other contexts. All calculations are rigorously rendered on an accurately devised time discretization scheme in which the limit passage is carried out via variational techniques.

Extension technique for functions of diffusion operators: a stochastic approach

It has recently been shown that complete Bernstein functions of the Laplace operator map the Dirichlet boundary condition of a related elliptic PDE to the Neumann boundary condition. The importance of this mapping consists in being able to convert problems involving non-local operators, like fractional Laplacians, into ones that only involve differential operators. We generalise this result to diffusion operators associated with stochastic differential equations, using a method which is entirely based on stochastic analysis.

Rough linear PDE’s with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion

We consider two related linear PDE’s perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.

Two-level schemes of Cauchy problem method for solving fractional powers of elliptic operators

In this paper we develop and investigate numerical algorithms for solving equations of the first order for the fractional powers of discrete elliptic operators: Aαy=φ, 0<α<1, for φ∈Vh with Vh a finite element or discrete approximation space. The main goal is to consider two different approaches to construct efficient high order time stepping schemes for the implementation of the Cauchy problem method (Vabishchevich, 2015). The first approach is based on a solution of the related time-dependent PDE, and the second approach is based on diagonal Padé approximations to (1+x)−α when the relation of the solution on two times levels is used. The second and fourth order approximations are constructed by both approaches. In order to increase the accuracy of approximations the graded time grid is constructed which compensates the singular behavior of the solution for t close to 0. Results of numerical experiments are presented, they agree well with the theoretical results.

A Feynman–Kac result via Markov BSDEs with generalised drivers

In this paper, we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman–Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE. Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process.

Profile of a touch-down solution to a nonlocal MEMS model

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on [Formula: see text] : [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] and [Formula: see text]. In this work, we have succeeded to construct a solution which quenches in finite time [Formula: see text] only at one interior point [Formula: see text]. In particular, we give a description of the quenching behavior according to the following final profile [Formula: see text] The construction relies on some connections between the quenching phenomenon and the blowup phenomenon. More precisely, we change our problem to the construction of a blowup solution for a related PDE and describe its behavior. The method is inspired by the work of Merle and Zaag [Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] with a suitable modification. In addition to that, the proof relies on two main steps: A reduction to a finite-dimensional problem and a topological argument based on index theory. The main difficulty and novelty of this work is that we handle the nonlocal integral term in the above equation. The interpretation of the finite-dimensional parameters in terms of the blowup point and the blowup time allows to derive the stability of the constructed solution with respect to initial data.

Application of Free Energy Perturbation (FEP+) to Understanding Ligand Selectivity: A Case Study to Assess Selectivity Between Pairs of Phosphodiesterases (PDE's).

Cyclic nucleotide phosphodiesterases (PDE's) are metalloenzymes that play a key role in regulating the levels of the ubiquitous second messengers, cyclic adenosine monophosphate (cAMP) and cyclic guanosine monophosphate (cGMP). In humans, 11 PDE protein families mediate numerous biochemical pathways throughout the body and are effective drug targets for the treatment of diseases ranging from central nervous system disorders to heart and pulmonary diseases. PDE's also share a highly conserved catalytic site (about 50%), thus making the design of selective drug candidates very challenging with classical structure-based design approaches given also the lack of publicly available co-crystal structures of pairs of PDE's with consistent biological data to be compared, as we show in our work. In this retrospective study, we apply free energy perturbation (FEP+) to predict the selectivity of inhibitors that bind two pairs of closely related PDE families: PDE9/1 and PDE5/6 where only 1 co-crystal structure per pair is publicly available. As another challenge, the p Ka of the PDE5/6 inhibitor is close to the experimental pH, making unclear the exact protonation state that should be used in the computational workflow. We demonstrate that running FEP+ on homology models constructed for these metalloenzymes accurately reproduces experimentally observed selectivity profiles also addressing the unclear protonation state to be used during computation with our recently developed p Ka-correction method. Based on these data, we conclude that FEP+ is a robust method for prediction of selectivity for this target class and may be helpful to address related lead optimization challenges in drug discovery.

From visco to perfect plasticity in thermoviscoelastic materials

AbstractWe consider a thermodynamically consistent model for thermoviscoplasticity. For the related PDE system, coupling the heat equation for the absolute temperature, the momentum balance with viscosity and inertia for the displacement variable, and the flow rule for the plastic strain, we propose two weak solvability concepts, ‘entropic’ and ‘weak energy’ solutions, where the highly nonlinear heat equation is suitably formulated. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Furthermore, we study the asymptotic behavior of weak energy solutions as the rate of the external data becomes slower and slower, which amounts to taking the vanishing‐viscosity and inertia limit of the system. We prove their convergence to a global energetic solution to the Prandtl‐Reuss model for perfect plasticity, whose evolution is ‘energetically’ coupled to that of the (spatially constant) limiting temperature.

Global Existence Results for Viscoplasticity at Finite Strain

We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable. The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.

Lie symmetry analysis, conservation laws and analytical solutions for the constant astigmatism equation

In this paper, the constant astigmatism equation is investigated, which finds numerous applications in geometry and physics describing surfaces of constant astigmatism. Utilizing a set of non-singular local multipliers, we present a set of local conservation laws for the equation, based on which, we further find a tree of nonlocally related PDE systems. Moreover, by considering these nonlocally related PDE systems, we systematically present Lie symmetries, optimal systems and some interesting analytical solutions of the constant astigmatism equation. Here it is the first time to investigate the nonlocally related PDE systems for this model, which can be used to expand the solution space of the given PDE system. Finally, by using its local conservation laws and variational principle, we obtain four kinds of Lagrangians.

Smoothing properties of McKean\u2013Vlasov SDEs

In this article, we develop integration by parts formulae on Wiener space for solutions of SDEs with general McKean–Vlasov interaction and uniformly elliptic coefficients. These integration by parts formulae hold both for derivatives with respect to a real variable and derivatives with respect to a measure understood in the sense of Lions. They allows us to prove the existence of a classical solution to a related PDE with irregular terminal condition. We also develop bounds for the derivatives of the density of the solutions of McKean–Vlasov SDEs.

Representations of Gaussian Random Fields and Approximation of Elliptic PDEs with Lognormal Coefficients

Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence \(y=(y_j)_{j\ge 1}\) of scalar random variables. One may then apply high-dimensional approximation methods to the solution map \(y\mapsto u(y)\). Although Karhunen–Loeve representations are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that multilevel-type expansions may yield better approximation rates. Motivated by these results, we construct wavelet-type representations of stationary Gaussian random fields defined on arbitrary bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approximation results for the related elliptic PDE which outperform those achievable when using Karhunen–Loeve representations. Our construction is based on a periodic extension of the stationary random field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Matern covariances. The proposed periodic continuation technique has other relevant applications such as fast simulation of trajectories. It can be regarded as a continuous analog of circulant embedding techniques introduced for Toeplitz matrices. One of its specific features is that the rate of decay of the eigenvalues of the covariance operator of the periodized process provably matches that of the Fourier transform of the covariance function of the original process.

Bohr's Correspondence Principle for the Renormalized Nelson Model

In the mid Sixties Edward Nelson proved the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field. Since then it is thought, despite the renormalization procedure involved in the construction, that the quantum dynamics should be governed in the classical limit by a Schr\"odinger-Klein-Gordon system with Yukawa coupling. In the present paper we prove this fact in the form of a Bohr correspondence principle. Besides, our result enlighten the nature of the renormalization method employed in this model which we interpret as a strategy that allows to put the related classical Hamiltonian PDE in a normal form suitable for a canonical quantization.

A modified multi-grid algorithm for a novel variational model to remove multiplicative noise

This paper proposes a novel variational model and a fast algorithm for its numerical approximation to remove multiplicative noise from digital images. By applying a maximum a posteriori (MAP), we obtained a strictly convex objective functional whose minimization leads to non-linear partial differential equations. As a result, developing a fast numerical scheme is difficult because of the high nonlinearity and stiffness of the associated Euler-Lagrange equation and standard unilevel iterative methods are not appropriate. To this end, we develop an efficient non-linear multi-grid algorithm with an improved smoother. We also discuss a local Fourier analysis of the associated smoothers which leads to a new and more effective smoother. Experimental results using both synthetic and realistic images, illustrate advantages of our proposed model in visual improvement as well as an increase in the peak signal-to-noise ratio over comparing to related recent corresponding PDE methods. We compare numerical results of new multigrid algorithm via modified smoother with traditional time marching schemes and with multigrid method via (local and global) fixed point smoother as well.

Generalized Orlicz spaces and related PDE

We prove the boundedness of the maximal operator in generalized Orlicz spaces defined on subsets of Rn. The proof is based on an extension result for Φ-functions. We study generalized Sobolev–Orlicz spaces and establish density of smooth functions and the Poincaré inequality. As applications we establish the existence of solutions of the φ-Laplace equation with zero and non-zero right-hand side. Further, we systematize assumptions for Φ-functions and prove several basic tools needed for the study of differential equations of generalized Orlicz growth.

Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems.

From an adhesive to a brittle delamination model in thermo-visco-elasticity

We address the analysis of a model for brittle delamination of two visco-elastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques.