Degenerate Equations Research Articles

The Onsager-Machlup Action Functional for Degenerate McKean-Vlasov Stochastic Differential Equations

The purpose of this paper is to investigate the existence of the Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations. To this end, we first derive Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations with constant diffusion in a broad set of norms by Girsanov transformation, some conditioned exponential inequalities and It $\mathrm{\hat{o}}$ formulas for distribution dependent functional. Then an example is given to illustrate our results.

Stability for the 2D magnetic Bénard system with partial dissipation and thermal damping near an equilibrium

In this paper, we focus on the Boussinesq equations with magnetohydrodynamics convection, in which the temperature obeys degenerate damped wave equations. Under the influence of an equilibrium, we prove that the 2D magnetic Bénard system remains stable in the whole space . In particular, we obtain the decay rate and large‐time behavior. Without the coupling, the 2D MHD system with only vertical dissipation is not known to be stable. It is revealed that both the thermal damping and the horizontal background magnetic field are indispensable.

Stabilization for degenerate equations with drift and small singular term

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

Simultaneous identification of the initial data in a degenerate and singular hyperbolic wave equation

Simultaneous identification of the initial data in a degenerate and singular hyperbolic wave equation

Degenerate Parabolic Equation with Zero-flux Boundary Condition and its Approximations

We study a degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The aim of this paper is to prove convergence of numerical approximate solutions towards the unique entropy solution. We propose an implicit finite volume scheme on admissible mesh. We establish fundamental estimates and prove that the approximate solution converge towards an entropy-process solution. Contrarily to the case of Dirichlet condition, in zero-flux problem unnatural boundary regularity of the flux is required to establish that entropyprocess solution is the unique entropy solution. In the study of well-posedness of the problem, tools of nonlinear semigroup theory (stationary, mild and integral solutions) were used in order to overcome this difficulty. Indeed, in some situations including the one-dimensional setting, solutions of the stationary problem enjoy additional boundary regularity. Here, similar arguments are developed based on the new notion of integral-process solution that we introduce for this purpose.

Self-Similar Solutions of a Multidimensional Degenerate Partial Differential Equation of the Third Order

When studying the boundary value problems’ solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order.

Maximum principle for non-uniformly parabolic equations and applications

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.

Correctness of the mixed problem for degenerate multidimensional elliptic-parabolic equations

The initial-boundary value problem (Dirichlet problem) for general elliptic-parabolic equations of second order was first posed by G. Fichera. Further investigation of this problem was carried out in the monograph by O.A. Oleinik and E.V. Radkevich and the works by V.N. Vragov. In these works, the authors examined mixed problems for degenerate multidimensional elliptic equations. The articles by S.A. Aldashev focused on the correctness (in the sense of uniqueness of solvability) of the Dirichlet problem in a cylindrical domain for multidimensional elliptic-parabolic equations. A mixed problem for these equations has not been studied. In this paper, the authors demonstrate the uniqueness of solvability and obtain an explicit representation of the classical solution of the mixed problem for degenerate multidimensional elliptic-parabolic equations. The proposed method allows reducing the problem under study to a mixed problem for a degenerate multidimensional elliptic equation examined by S.A. Aldashev.

Neumann boundary control for degenerate wave equations in time-varying domains

Degenerate wave equation with Neumann boundary control in time-varying domains is considered. The fixed endpoint has a Dirichlet-type boundary condition, a Neumann boundary control applied to the moving end. By using the Hilbert Uniqueness Method, we establish the exact boundary controllability for the degenerate wave equation and obtain estimation for the control time.

Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front.

On weak existence of solutions of degenerate McKean-Vlasov equations

On weak existence of solutions of degenerate McKean-Vlasov equations

Stability for some classes of degenerate nonlinear hyperbolic equations with time delay

We consider several classes of degenerate hyperbolic equations involving delay terms and suitable nonlinearities. The idea is to rewrite the problems in an abstract way and, using semigroup theory and energy method, we study well-posedness and stability. Moreover, some illustrative examples are given.

The Cauchy problem for a class of linear degenerate evolution equations on the torus

We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator P(t,Dx,Dt)=Dt−a2(t)Δx+∑j=1Na1,j(t)Dxj+a0(t), with T>0, t∈[0,T] and a2,a1,1,…,a1,N,a0∈C([0,T];C). Using Fourier analysis techniques, we obtain a complete characterization for the well-posedness of a class of degenerate initial-value problems in the Sobolev, Smooth, Gevrey and Real-Analytic frameworks.

New Algorithm for Constructing Asymptotic Solutions of Singularly Perturbed Optimal Control Problems with Intersecting Trajectories of the Degenerate State Equation

New Algorithm for Constructing Asymptotic Solutions of Singularly Perturbed Optimal Control Problems with Intersecting Trajectories of the Degenerate State Equation

Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.

The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect

The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect

Study of a degenerate non-elliptic equation to model plasma heating

Study of a degenerate non-elliptic equation to model plasma heating

A Nonlocal Problem for a Degenerate Equation of Elliptic Type with a Singular Coefficient

A Nonlocal Problem for a Degenerate Equation of Elliptic Type with a Singular Coefficient

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient