Continuous Dependence Estimates Research Articles

A continuous dependence estimate for viscous Hamilton–Jacobi equations on networks with applications

A continuous dependence estimate for viscous Hamilton–Jacobi equations on networks with applications

Kolmogorov continuity and stability of sample paths of entropy solutions of stochastic conservation laws

This paper is concerned with sample paths and path-based properties of the entropy solution to conservation laws with stochastic forcing. We derive a series of uniform maximal-type estimates for the viscous perturbation and establish the existence of stochastic entropy solution that has Hölder continuous sample paths. This information is then carefully choreographed with Kružkov’s technique to obtain stronger continuous dependence estimates, based on the nonlinearities, for the sample paths of the solutions. Finally, convergence of sample paths is established for vanishing viscosity approximation along with an explicit rate of convergence.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

A Cahn–Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials

A system with equation and dynamic boundary condition of Cahn–Hilliard type is considered. This system comes from a derivation performed in Liu–Wu (Arch. Ration. Mech. Anal., 233:167–247, 2019) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter.

Well-Posedness and Global Attractors for Viscous Fractional Cahn–Hilliard Equations with Memory

We examine a viscous Cahn–Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the nonlinear potential allow for arbitrary polynomial growth.

The Cahn--Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn-Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward dynamic boundary condition at the limit. This is done in a very general setting, with nonlinear terms admitting maximal monotone graphs both in the bulk and on the boundary. The two graphs are related by a growth condition, with the boundary graph that dominates the other one. It turns out that in the limiting procedure the solution of the problem looses some regularity and the limit equation has to be properly interpreted in the sense of a subdifferential inclusion. However, the limit problem is still well-posed since a continuous dependence estimate can be proved. Moreover, in the case when the two graphs exhibit the same growth, it is shown that the solution enjoys more regularity and the boundary condition holds almost everywhere. An error estimate can also be shown, for a suitable order of the diffusion parameter.

Nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing

We are concerned with nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing, which are heterogeneous (i.e., not space-translational invariant). A unified framework is established for the continuous dependence estimates, fractional BV regularity estimates, and well-posedness for stochastic kinetic solutions of the nonlinear stochastic degenerate parabolic-hyperbolic equation. In particular, we establish the well-posedness of the nonlinear stochastic equation in Lp∩Nκ,1 for p∈[1,∞) and the κ–Nikolskii space Nκ,1 with κ∈(0,1], and the L1–continuous dependence of the stochastic kinetic solutions not only on the initial data, but also on the degenerate diffusion matrix function, the flux function, and the multiplicative noise function involved in the nonlinear equation.

A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method “à la Kružkov” for more general equations where the noise coefficient may depend explicitly on the spatial variable.

Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

On the quasi-static approximation in the initial boundary value problem of linearised elastodynamics

Continuous data dependence estimates are employed to rigorously derive conditions that validate the quasi-static approximation for the initial homogeneous boundary value problem in the theory of small elastic deformations superposed upon large elastic deformations. This theory imposes no sign-definite assumptions on the linearised elastic moduli and in consequence the requisite estimates are established using methods principally motivated by known Lagrange identity arguments.

Well-posedness for the coupling of a random heat equation with a multiplicative stochastic Barenblatt equation

In this contribution, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a random heat equation coupled with a Barenblatt’s type equation with a multiplicative stochastic force in the sense of Itô. In a first step, we establish well-posedness in the case of an additive noise through a semi-implicit time discretization of the system. In a second step, the derivation of continuous dependence estimates of the solution with respect to the data allows us to show the desired existence and uniqueness result for the multiplicative case.

Lipschitz stable determination of small conductivity inclusions in a semilinear equation from boundary data

We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [ 4 ]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.

The Cauchy problem for fractional conservation laws driven by Lévy noise

In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by Lévy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that the Lévy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. Furthermore, we establish a result on vanishing non-local regularization of scalar stochastic conservation laws.

Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an n -dimensional Lévy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub- and supersolutions which we use as “barrier functions” to prove Hölder continuity in the time variable, and, through Perron's method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) [5] and of Lundström, Nyström and Olofsson (2014) [21] , [22] to hold for more general systems of equations.

Pointwise rates of convergence for the Oliker–Prussner method for the Monge–Ampère equation

We study the Oliker-Prussner method exploiting its geometric nature. We derive discrete stability and continuous dependence estimates in the max-norm by using a discrete Alexandroff estimate and the Brunn-Minkowski inequality. We show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain; nodes still conform to the domain boundary. This gives a suitable notion of operator consistency which, combined with stability, leads to pointwise rates of convergence for classical and non-classical solutions of the Monge-Amp\`{e}re equation.

On a Class of Conserved Phase Field Systems with a Maximal Monotone Perturbation

We prove existence and regularity for the solutions to a Cahn–Hilliard system describing the phenomenon of phase separation for a material contained in a bounded and regular domain. Since the first equation of the system is perturbed by the presence of an additional maximal monotone operator, we show our results using suitable regularization of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments. Next, under further assumptions, we deduce a continuous dependence estimate whence the uniqueness property is also achieved. Then, we consider the related sliding mode control (SMC) problem and show that the chosen SMC law forces a suitable linear combination of the temperature and the phase to reach a given (space-dependent) value within finite time.

Continuous dependence estimate for a degenerate parabolic–hyperbolic equation with Lévy noise

In this article, we are concerned with a multidimensional degenerate parabolic–hyperbolic equation driven by Levy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Levy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficients depend on both the solution and spatial variable.

The time differential three-phase-lag heat conduction model: Thermodynamic compatibility and continuous dependence

This paper deals with the time differential three-phase-lag heat transfer model aiming, at first, to identify the restrictions that make it thermodynamically consistent. The model is thus reformulated by means of the fading memory theory, in which the heat flux vector depends on the history of the thermal displacement gradient: the thermodynamic principles are then applied to obtain suitable restrictions involving the delay times. Consistently with the thermodynamic restrictions just obtained, a first result about the continuous dependence of the solutions with respect to the given initial data and to the supply term is established for the related initial boundary value problems. Subsequently, to provide a more comprehensive review of the problem, a further continuous dependence estimate is proved, this time conveniently relaxing the hypotheses about the above-said thermodynamic restrictions. This last estimate allows the solutions to grow exponentially in time and so to have asymptotic instability.

A stabilized nonconforming finite element method for the elliptic Cauchy problem

In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data on the estimates is investigated. The recently derived framework from \cite{Bu13,Bu14} is extended to include the case of nonconforming approximation spaces and we show that the use of such spaces allows us to reduce the amount of stabilization necessary for convergence, even in the case of ill-posed problems.

Continuous dependence estimate for conservation laws with Lévy noise

We are concerned with multidimensional stochastic balance laws driven by Lévy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise only depends on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable.