Involving Gradient Terms Research Articles

A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

AbstractIn this article we study local and global properties of positive solutions of − Δmu = ∣u∣p−1u+M∣∇u∣q in a domain Ω of ℝN, with m > 1, p, q > 0 and M ∈ ℝ. Following some ideas used in [7, 8], and by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin’s classical results.

A deep branching solver for fully nonlinear partial differential equations

We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any orders, by combining the use of neural networks with a Monte Carlo branching algorithm. In comparison with other deep learning PDE solvers, it also allows us to check the consistency of the learned neural network function. Numerical experiments presented show that this algorithm can outperform deep learning approaches based on backward stochastic differential equations or the Galerkin method, and provide solution estimates that are not obtained by those methods in fully nonlinear examples.

Sharp Liouville type results for semilinear elliptic inequalities involving gradient terms on weighted graphs

We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms$ \Delta u+u^p\left|\nabla u\right|^q\leq0, $on weighted graphs, where $ (p,q)\in\mathbb{R}^2 $. We give a complete classification of $ (p,q) $ under which sharp volume growth assumptions are established.

Singular quasilinear elliptic systems with gradient dependence

In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder’s fixed point theorem.

The cost of approximate controllability of heat equation with general dynamical boundary conditions

We consider the heat equation with dynamic boundary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.

Singular quasilinear elliptic systems involving gradient terms

In this paper we establish the existence of at least one smooth positive solution for a singular quasilinear elliptic system involving gradient terms. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.

Existence results and a priori estimates for solutions of quasilinear problems with gradient terms

In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in \(\mathbb{R}^N\) with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of \(u\) and by removing the growth condition for the nonlinearity \(f\) at \(u=0\).

Galilean-invariant preconditioned central-moment lattice Boltzmann method without cubic velocity errors for efficient steady flow simulations.

Lattice Boltzmann (LB) models used for the computation of fluid flows represented by the Navier-Stokes (NS) equations on standard lattices can lead to non-Galilean-invariant (GI) viscous stress involving cubic velocity errors. This arises from the dependence of their third-order diagonal moments on the first-order moments for standard lattices, and strategies have recently been introduced to restore Galilean invariance without such errors using a modified collision operator involving corrections to either the relaxation times or the moment equilibria. Convergence acceleration in the simulation of steady flows can be achieved by solving the preconditioned NS equations, which contain a preconditioning parameter that can be used to tune the effective sound speed, and thereby alleviating the numerical stiffness. In the present paper, we present a GI formulation of the preconditioned cascaded central-moment LB method used to solve the preconditioned NS equations, which is free of cubic velocity errors on a standard lattice, for steady flows. A Chapman-Enskog analysis reveals the structure of the spurious non-GI defect terms and it is demonstrated that the anisotropy of the resulting viscous stress is dependent on the preconditioning parameter, in addition to the fluid velocity. It is shown that partial correction to eliminate the cubic velocity defects is achieved by scaling the cubic velocity terms in the off-diagonal third-order moment equilibria with the square of the preconditioning parameter. Furthermore, we develop additional corrections based on the extended moment equilibria involving gradient terms with coefficients dependent locally on the fluid velocity and the preconditioning parameter. Such parameter dependent corrections eliminate the remaining truncation errors arising from the degeneracy of the diagonal third-order moments and fully restore Galilean invariance without cubic defects for the preconditioned LB scheme on a standard lattice. Several conclusions are drawn from the analysis of the structure of the non-GI errors and the associated corrections, with particular emphasis on their dependence on the preconditioning parameter. The GI preconditioned central-moment LB method is validated for a number of complex flow benchmark problems and its effectiveness to achieve convergence acceleration and improvement in accuracy is demonstrated.

Derivation of a hydrodynamic theory for mesoscale dynamics in microswimmer suspensions.

In this paper, we systematically derive a fourth-order continuum theory capable of reproducing mesoscale turbulence in a three-dimensional suspension of microswimmers. We start from overdamped Langevin equations for a generic microscopic model (pushers or pullers), which include hydrodynamic interactions on both small length scales (polar alignment of neighboring swimmers) and large length scales, where the solvent flow interacts with the order parameter field. The flow field is determined via the Stokes equation supplemented by an ansatz for the stress tensor. In addition to hydrodynamic interactions, we allow for nematic pair interactions stemming from excluded-volume effects. The results here substantially extend and generalize earlier findings [S. Heidenreich et al., Phys. Rev. E 94, 020601 (2016)2470-004510.1103/PhysRevE.94.020601], in which we derived a two-dimensional hydrodynamic theory. From the corresponding mean-field Fokker-Planck equation combined with a self-consistent closure scheme, we derive nonlinear field equations for the polar and the nematic order parameter, involving gradient terms of up to fourth order. We find that the effective microswimmer dynamics depends on the coupling between solvent flow and orientational order. For very weak coupling corresponding to a high viscosity of the suspension, the dynamics of mesoscale turbulence can be described by a simplified model containing only an effective microswimmer velocity.

A priori bounds and existence of solutions for some nonlocal elliptic problems

In this paper we show existence of solutions for some elliptic problems with nonlocal diffusion by means of nonvariational tools. Our proof is based on the use of topological degree, which requires a priori bounds for the solutions. We obtain the a priori bounds by adapting the classical scaling method of Gidas and Spruck. We also deal with problems involving gradient terms.

Solutions for a Class of Nonperiodic Superquadratic Hamiltonian Elliptic Systems Involving Gradient Terms

In the present paper, we consider the following Hamiltonian elliptic system HES: -Δu+bx·∇u+Vxu=Hvx,u,v, x∈RN, -Δv-bx·∇v+Vxv=Hux,u,v, x∈RN. A new existence result of nontrivial solutions is obtained for the system HES via variational methods for strongly indefinite problems, which generalizes some known results in the literatures.

Liouville type theorems for elliptic systems involving gradient terms

The objective of this work is to obtain Liouville type theorems for positive supersolutions of the elliptic system{−Δu+|∇u|=λvp,−Δv+|∇v|=μus, in exterior domains of RN, where p,s>0 and λ,μ>0 are parameters. We prove that positive supersolutions do not exist in some ranges of the parameters which turn out to be optimal.

Null controllability with constraints on the state for nonlinear heat equations

We consider the null controllability problem with a finite number of constraints on the state for a nonlinear heat equation involving gradient terms in a bounded domain of . The control is distributed along a bounded subset of the domain and the nonlinearity is assumed to be of class and globally Lipschitz. Interpreting each constraint in terms of the notion of adjoint state, we transform the linearized problem into an equivalent null controllability problem with constraint on the control. Using a Carleman inequality adapted to the constraint, we prove first the null controllability of the linearized problem. Then, by a fixed-point method, we show that the same result holds when the nonlinearity is of class and globally Lipschitz.

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of R N . The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem.

Insensitizing controls for a nonlinear parabolic equation with a nonlinear term involving the state and the gradient in unbounded domains

In this paper, we consider the existence of insensitizing control for a semilinear heat equation involving gradient terms in unbounded domain Ω . In this case, we prove the existence of controls insensitizing the L 2 -norm of the observation of the solution in an open subset of the domain. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

The cost of approximate controllability for semilinear heat equations

We consider the semilinear heat equation with globally Lipschitz non-linearity involving gradient terms in a bounded domain of ℝn. In this paper, we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. The methods we used combine global Carleman estimates, the variational approach to approximate controllability and Schauder’s fixed point theorem.

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this paper, we prove the controllability of a quasi-linear heat equation involving gradient terms with Fourier boundary conditions in a bounded domain of ℝ N . The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of R N . We first consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F( y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L 2-norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

Approximate controllability for the semilinear heat equation in R N involving gradient terms

Approximate controllability for the semilinear heat equation in R N involving gradient terms

Approximate controllability for the semilinear heat equation in R N involving gradient terms

We prove the approximate controllability of the semilinear heat equation in R N , when the nonlinear term is globally Lipschitz and depends both on the state u and its spatial gradientu. The approximate controllability is viewed as the limit of a sequence of optimal control problems. In order to avoid the difficulties related to the lack of compactness of the Sobolev embeddings, we work with the similarity variables and use weighted Sobolev spaces.