Non-regular Coefficients Research Articles

Hierarchical Control for the Oldroyd Equation in Memoriam to Professor Luiz Adauto Medeiros.

This manuscript deals with a hierarchical control problem for Oldroyd equation under the Stackelberg-Nash strategy. The Oldroyd equation model is defined by non-regular coefficients, that is, they are bounded measurable functions. We assume that we can act in the dynamic of the system by a hierarchy of controls, where one main control (the leader) and several additional secondary control (the followers) act in order to accomplish their given tasks: controllability for the leader and optimization for followers. We obtain the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability with respect to the leader control, and the optimality system for leader control.

Exponential almost sure synchronization of one-dimensional diffusions with nonregular coefficients

We study the asymptotic behavior of a real-valued diffusion whose nonregular drift is given as a sum of a dissipative term and a bounded measurable one. We prove that two trajectories of that diffusion converge almost sure to one another at an exponential explicit rate as soon as the dissipative coefficient is large enough. A similar result in Lp is obtained.

Initial-boundary value problem for stochastic transport equations

This paper concerns the Dirichlet initial-boundary value problem for stochastic transport equations with non-regular coefficients. First, the existence and uniqueness of the strong stochastic traces is proved. The existence of weak solutions relies on the strong stochastic traces, and also on the passage from the Stratonovich into Ito’s formulation for bounded domains. Moreover, the uniqueness is established without the divergence of the drift vector field bounded from below.

On the Asymptotics of Solutions of a Singular nth-Order Differential Equation with Nonregular Coefficients

On the Asymptotics of Solutions of a Singular nth-Order Differential Equation with Nonregular Coefficients

Regularization by Noise in One-Dimensional Continuity Equation

A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which the classical DiPerna-Lions-Ambrosio theory of uniqueness of distributional solutions does not apply. We solve partially the open problem that is the case when the vector-field has random dependence. In addition, we prove a stability result for the solutions.

On the Asymptotics of Solutions of a Singular $n$th-Order Differential Equation with Nonregular Coefficients

On the Asymptotics of Solutions of a Singular $n$th-Order Differential Equation with Nonregular Coefficients

On hyperbolic equations and systems with non-regular time dependent coefficients

In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than Hölder, namely bounded coefficients. As for second order equations in [14] we prove that such equations admit a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter. Classical solutions are recovered as limit of very weak solutions. Finally, by using a reduction to block Sylvester form we conclude that any first order hyperbolic system with non-regular coefficients is solvable in the very weak sense.

Renormalized Solutions for Stochastic Transport Equations and the Regularization by Bilinear Multiplicative Noise

A linear stochastic transport equation with non-regular coefficients is considered. Under the same assumption of the deterministic theory, all weak L ∞-solutions are renormalized. But then, if the noise is non-degenerate, uniqueness of weak L ∞-solutions does not require essential new assumptions, opposite to the deterministic case where for instance the divergence of the drift is asked to be bounded. The proof gives a new explanation why bilinear multiplicative noise may have a regularizing effect.

Null controllability of the heat equation with boundary Fourier conditions: the linear case

In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂y ∂n + βy = 0. We consider distributed controls with support in a small set and nonregular coefficients β = β(x, t). For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions.

The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

In this paper we consider the Cauchy problem for the equation\(\partial ^2 u\left( {t,x} \right)/\partial t^2 = - \sum\nolimits_{j,k = 1}^n {D_j } \left( {a_{jk} \left( {t,x} \right)D_k u\left( {t,x} \right)} \right) + f\left( {t,x} \right)\), where the matrix {ajk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs<q/(q−1).

Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem

We deal with the Cauchy problem for a strictly hyperbolic second-order operator with non-regular coefficients in the time variable. It is well-known that the problem is well-posed in L 2 in case of Lipschitz continuous coefficients and that the log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here, we prove that any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification: modulus of continuity → loss of derivatives is sharp Lipschitz → no loss , intermediate → arbitrary small loss , log - Lipschitz → finite loss .

Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C ∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C ∞ . In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.

Green function for X-elliptic operators

We study the Green function associated to second order X-elliptic operators with non-regular coefficients. An existence and uniqueness theorem is given, along with some local estimations and a representation formula for the solution of the Dirichlet problem.

Operators of p-evolution with nonregular coefficients in the time variable

We study the Cauchy problem for a class of p-evolution operators P( t, x, D t , D x ) in [0,T]× R n, p>1 , with less than C 1 coefficients with respect to the time variable. According to Lipschitz, log-lipschitz or Hölder regularity we find well-posedness in Sobolev spaces or in Gevrey classes.

Operators of p-evolution with nonregular coefficients in the time variable

We study the Cauchy problem for a class of p -evolution operators P ( t , x , D t , D x ) in [0,T]× R n , p>1 , with less than C 1 coefficients with respect to the time variable. According to Lipschitz, log-lipschitz or Hölder regularity we find well-posedness in Sobolev spaces or in Gevrey classes.

On weakly hyperbolic operators with non-regular coefficients and finite order degeneration

We prove that the Cauchy problem for a class of weakly hyperbolic equations satisfying a condition of finite order degeneration and having non-Lipschitz-continuous coefficients is well-posed in Gevrey spaces.

Gevrey-well-posedness for weakly hyperbolic operators with non-regular coefficients

We prove that the Cauchy problem for a class of weakly hyperbolic equations with non-Lipschitz coefficients is well-posed in Gevrey spaces.

On Polynomial Mixing for SDEs with a Gradient-Type Drift

Convergence rate to an invariant distribution and rate of mixing is estimated for a class of stochastic differential equations with nonregular coefficients.

Stochastic differential equations with nonregular coefficients

Stochastic differential equations with nonregular coefficients