Discretizing a system of parabolic nonlinear equations with a discontinuous solution

A new initial— boundary-value problem is posed for a system of two nonlinear parabolic equations with convective transport terms that allow a discontinuity in the solution. Discretization on classes of discontinuous functions is proposed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 33–42, 1992.     

Existence of a stationary symmetric solution of the two-dimensional Navier-Stokes equations with a given head and with nonzero fluxes

We consider a stationary boundary value problem for the Navier-Stokes equations of a homogeneous incompressible fluid in a two-dimensional bounded domain with boundary consisting of connected components Γ _i . On each part Γ _i , we specify the tangent component of the flow velocity vector, the total flow head (up to an additive constant), and the fluid flux through Γ _i . For the case in which the domain and the original data are symmetric around some line, we prove the existence of a solution of the problem with such a symmetry. We also present some results on the solvability in the nonsymmetric case.     

Existence and asymptotic stability of periodic solution for evolution equations with delays

In this paper, we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a Hilbert space H

     

Existence of solutions for a system of diffusion equations with spectrum point zero

In this paper, we consider the existence and multiplicity of homoclinic type solutions to a system of diffusion equations with spectrum point zero. By using some recent critical point theorems for strongly indefinite problems, we obtain at least one nontrivial solution and also infinitely many solutions.     

Existence of the stationary solution of a Rayleigh-type equation

A fluid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thin boundary layer (“lower deck”) and a classical Prandtl boundary layer (“upper deck”). The aim of this paper is to prove the existence and uniqueness of the stationary solution of a Rayleigh-type equation, which describes oscillations of the vertical velocity component in the classical boundary layer.     

Existence of the mild solution for some fractional differential equations with nonlocal conditions

We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D ^q x(t)=Ax(t)+t ⁿ f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ⁺, x(0)+g(x)=x ₀, where 0<q<1, A is the infinitesimal generator of a C ₀-semigroup of bounded linear operators on a Banach space X.     

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

This paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained.     

Existence and stability of planar diffusion waves for 2-D Euler equations with damping

To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L₂ convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L₂ norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the Lp estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties.     

Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal

This paper is concerned with a nonlocal evolution equation which is used to model the spatial dispersal of organisms. We study the existence, uniqueness and stability of the positive steady solution for this nonlocal evolution equation under general conditions. The global dynamics are also investigated and a trichotomy of the global asymptotics is established.     

Existence of solution of a finite volume scheme preserving maximum principle for diffusion equations

In this article, a cell‐centered finite volume scheme preserving maximum principle for diffusion equations with scalar coefficients is developed. The construction of the scheme consists of three steps: at first the discrete normal flux is obtained by a linear combination of two single‐sided fluxes, then the tangential term of the normal flux is modified by using a nonlinear combination of two single‐sided tangential fluxes, finally the auxiliary unknowns in the tangential fluxes are calculated by the convex combinations of the cell‐centered unknowns. It is proved that this nonlinear scheme satisfies the discrete maximum principle (DMP). Moreover, the existence of a solution of the nonlinear scheme is proved by using the Brouwer's fixed point theorem and the bounded estimates. Numerical experiments are presented to show that the scheme not only satisfies DMP, but also obtains the second‐order accuracy and conservation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 80–96, 2018     

Existence and stability of bounded solutions for a system of parabolic equations

In this paper we study the existence and the stability of bounded solutions of the following non-linear system of parabolic equations with homogeneous Dirichlet boundary conditions:

     

On the two-dimensional tidal dynamics system: stationary solution and stability

ABSTRACT

In this work, we consider the two-dimensional stationary and non-stationary tidal dynamic equations and examine the asymptotic behavior of the stationary solution. We prove the existence and uniqueness of weak and strong solutions of the stationary tidal dynamic equations in bounded domains using compactness arguments. Using maximal monotonicity property of the linear and nonlinear operators, we also establish that the solvability results are even valid in unbounded domains. Later, we obtain a uniform Lyapunov stability of the steady state solution. Finally, we remark that the stationary solution is exponentially stable if we add a suitable dissipative term in the equation corresponding to the deviations of free surface with respect to the ocean bottom. This exponential stability helps us to ensure the mass conservation of the modified system, if we choose the initial data of the modified system as stationary solution.