Existence and uniqueness of weak solutions for a system applied to image decomposition

In this paper, we prove the existence and uniqueness of weak solutions for a singular evolutionary system, which is deduced from a model for image decomposition combining staircase reduction and texture extraction. The main method we used is p-Laplace regularization. The numerical experimental result shows the efficiency of this kind of model.     

Existence and uniqueness of weak solutions for some singular evolutionary system in image denoising

In this paper, we prove the existence and uniqueness of weak solutions for some singular evolutionary system, which is deduced from one of the Chambolle–Lions denoising models. The main method we used is p‐Laplace regularization. Copyright © 2013 John Wiley & Sons, Ltd.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

Global existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity

We prove the global‐in‐time existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity, with initial data ψ₀,A₀ ∈ L³,u₀ ∈ L^{3 ∕ 2},u₀ ≥ 0 in three dimension and in two dimension. Copyright © 2014 John Wiley & Sons, Ltd.     

A note on existence and uniqueness of limit cycles for Liénard systems

We consider the Liénard equation and we give a sufficient condition to ensure existence and uniqueness of limit cycles. We compare our result with some other existing ones and we give some applications.     

Existence and uniqueness of solutions for a diffusion model of host-parasite dynamics

Milner and Patton (J. Comput. Appl. Math., in press) introduced earlier a new approach to modeling host-parasite dynamics through a convection-diffusion partial differential equation, which uses the parasite density as a continuous structure variable. A motivation for the model was presented there, as well as results from numerical simulations and comparisons with those from other models. However, no proof of existence or uniqueness of solutions to the new model proposed was included there. In the present work the authors deal with the well posedness of that model and they prove existence and uniqueness of solutions, as well as establishing some asymptotic results.     

EXISTENCE，UNIQUENESS AND PROPERTIES OF THE SOLUTIONS OF A DEGENERATE PARABOLIC EQUATION WITH DIFFUSION－ADVECTION－ABSORPTION

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Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

The solvability of initial value problems for nonlinear Langevin equation involving two fractional orders are discussed in this paper. An existence result for the solution is obtained using the Leray–Schauder nonlinear alternative. In addition, sufficient conditions for unique solution are established under the Banach contraction principle. The existence results for the initial value problems of nonlinear classical Langevin equation follow as a special case of our results.     

Existence and uniqueness of weak solutions for a non-uniformly parabolic equation

In this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation.     

Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations

By means of a monotone iterative technique, we establish the existence and uniqueness of the positive solutions for a class of higher conjugate-type fractional differential equation with one nonlocal term. In addition, the iterative sequences of solution and error estimation are also given. In particular, this model comes from economics, financial mathematics and other applied sciences, since the initial value of the iterative sequence can begin from an known function, this is simpler and helpful for computation.     

Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation

In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a fourth-order nonlinear parabolic equation.     

The existence,uniqueness, and regularity for an incompressible Newtonian flow with intrinsic degree of freedom

We consider the regularity and uniqueness of solution to the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid, taking into account internal degree of freedom. We first show there exist uniquely a local strong solution. Then we show this solution can be extend to the whole interval [0,T] if the velocity u, or its gradient ? u, or the pressure p belongs to some function class, which are similar with that of incompressible Navier–Stokes equations. Our result shows that the solution is unique in these classes, and that velocity field plays a more prominent role in the existence theory of strong solution than the angular velocity field. Finally, if the L^{3 ∕ 2}‐norm of the initial angular velocity vector and some homogeneous Besov norm of initial velocity field are small, then there exists uniquely a global strong solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order

In this paper, we prove some existence and uniqueness of mild solutions for a semilinear integrodifferential equation of fractional order with nonlocal initial conditions in α-norm. We assume that the linear part generates a noncompact analytic semigroup. Our results cover the cases that the nonlinearity F takes values in different spaces such as X,X_α and X_β, where α≤β(0,1). Finally, some practical consequences are also obtained.     

Existence and uniqueness of steady solutions to the magnetohydrodynamic equations of compressible flows

We establish the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary, under the assumption that the external force field is small.     

Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients

In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the Stroock-Varadhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for Ld-almost every initial condition x whenever existence and uniqueness is known at the PDE level in the L^∞-setting (and, conversely, if existence and uniqueness of martingale solutions is known for Ld-a.e. initial condition, then existence and uniqueness for the PDE holds). In the second part of the paper we consider situations where, on the one hand, no pointwise uniqueness result for the martingale problem is known and, on the other hand, well-posedness for the Fokker-Planck equation can be proved. Thus, the theory developed in the first part of the paper is applicable. In particular, we will study the Fokker-Planck equation in two somehow extreme situations: in the first one, assuming uniform ellipticity of the diffusion coefficients and Lipschitz regularity in time, we are able to prove existence and uniqueness in the L²-setting; in the second one we consider an additive noise and, assuming the drift b to have BV regularity and allowing the diffusion matrix a to be degenerate (also identically 0), we prove existence and uniqueness in the L^∞-setting. Therefore, in these two situations, our theory yields existence, uniqueness and stability results for martingale solutions.     

On existence and uniqueness of solutions to uncertain backward stochastic differential equations简

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.     

Dynamics of droplets in an agitated dispersion with multiple breakage. Part II: uniqueness and global existence

In Part I (see Math. Methods Appl. Sci.) a new model for the evolution of a system of droplets dispersed in an agitated liquid was presented. Our aim was to extend a previous version (see Proceedings of the Symposium Organized by the Sonderforschungsbereich 438 on the Occasion of Karl‐Heinz Hoffman's 60th Birthday, Lectures in Applied Mathematics, Springer: Berlin, 2000; 123–141) in order to describe the influence of each breakage mode. Here we complete the mathematical analysis to ensure the well posedness (in the sense of Hadamard) of the Cauchy problem for the main evolution equation. Copyright © 2005 John Wiley & Sons, Ltd.