Low Regularity Coefficients Research Articles

Interior transmission problems with coefficients of low regularity

Interior transmission problems with coefficients of low regularity

Heat kernel and gradient estimates for kinetic SDEs with low regularity coefficients

We establish heat kernel and gradient estimates for the density of kinetic degenerate Kolmogorov stochastic differential equations. Our results are established under somehow minimal assumptions that guarantee the SDE is weakly well posed.

Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity

We prove a formula representing solutions to parabolic Fokker–Planck–Kolmogorov equations with coefficients of low regularity. This formula is applied for proving the continuity of solution densities under broad assumptions and obtaining upper bounds for them. In the case of diffusion coefficients of class VMO$$_x$$, we show that the solution density is locally integrable to any power.

Integrability and continuity of solutions to double divergence form equations

We obtain sharp conditions for higher integrability and continuity of solutions to double divergence form second-order elliptic equations with coefficients of low regularity. In addition, we prove Harnack’s inequality in this case.

Lagrangian solutions to the Vlasov–Poisson system with [formula omitted] density

The recently developed theory of Lagrangian flows for transport equations with low regularity coefficients enables to consider non BV vector fields. We apply this theory to prove existence and stability of global Lagrangian solutions to the repulsive Vlasov–Poisson system with only integrable initial distribution function with finite energy. These solutions have a well-defined Lagrangian flow. An a priori estimate on the smallness of the superlevels of the flow in three dimensions is established in order to control the characteristics.

The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients

In this paper we study the well-posedness of the Cauchy problem for first order hyperbolic systems with constant multiplicities and with low regularity coefficients depending just on the time variable. We consider Zygmund and log-Zygmund type assumptions, and we prove well-posedness in H∞ respectively without loss and with finite loss of derivatives. The key to obtain the results is the construction of a suitable symmetrizer for our system, which allows us to recover energy estimates (with or without loss) for the hyperbolic operator under consideration. This can be achievied, in contrast with the classical case of systems with smooth (say Lipschitz) coefficients, by adding one step in the diagonalization process, and building the symmetrizer up to the second order.

Time-Dependent Loss of Derivatives for Hyperbolic Operators with Non Regular Coefficients

In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension N ≥ 1. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradifferential calculus with parameters will be the main tool to get energy estimates in Sobolev spaces and these estimates will present a time-dependent loss of derivatives.

A well-posed Cauchy problem for an evolution equation with coefficients of low regularity

In the hyperbolic Cauchy problem, the well-posedness in Sobolev spaces is strictly related to the modulus of continuity of the coefficients. This holds true for p-evolution equations with real characteristics (p=1 hyperbolic equations, p=2 vibrating plate and Schrödinger type models, …). We show that, for p⩾2, a lack of regularity in t can be balanced by a damping of the too fast oscillations as the space variable x→∞. This cannot happen in the hyperbolic case p=1 because of the finite speed of propagation.

On divergence-free drifts

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form ∂ t − Δ + b ⋅ ∇ resp. − Δ + b ⋅ ∇ with a divergence-free drift b. We prove the Liouville theorem and Harnack inequality when b ∈ L ∞ ( BMO − 1 ) resp. b ∈ BMO − 1 and provide a counterexample demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BMO skew-symmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the non-scale-invariant norm ‖ b ‖ L 1 . In three dimensions, on the other hand, bounded solutions with L 1 drifts may be discontinuous.

A parametrix construction for the wave equation with low regularity coefficients using a frame of Gaussians

We construct a frame of complex Gaussians for the space of L2 (Rn) functions. When propagated along bicharacteristics for the wave equation, the frame can be used to build a parametrix with suitable error terms. When the coefficients of the wave equation have more regularity, propagated frame functions become Gaussian beams.

Wave-type equations of low regularity

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second-order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

Addendum to “Concentration and Lack of Observability of Waves in Highly Heterogeneous Media”

In [1] we introduced a class of 1−d wave equations with rapidly oscillating Holder continuous coefficients for which the classical boundary observability property fails. We also established that these examples could be used to contradict Strichartz-type inequalities for the wave equation with low regularity coefficients. The object of this addendum is to further analyze this issue. As we will see, the argument in [1] only provides sharp counter-examples to the Strichartz estimates when the coefficient ρ belongs to L ∞. We carefully analyze these counter-examples for Holder continuous coefficients. We also give a new application of our construction which shows that some eigenfunction estimates for elliptic operators due to Sogge can fail when coefficients are not smooth enough.

Linear thermo-elasticity in non-smooth media

An alternative first order formulation of linear thermo-elasticity is presented, designed to deal with coefficients of low regularity as a model for a large class of media occupying arbitrary non-empty open sets. By fine-tuning the concepts of generalized derivatives a general existence, uniqueness and continuous dependence result is obtained in the framework of extrapolation spaces (Sobolev lattices). Copyright © 2005 John Wiley & Sons, Ltd.

Dispersion and Strichartz Inequalities for Schrödinger Equations with Singular Coefficients

In this paper we prove the global dispersion and the Strichartz inequalities for a class of one-dimensional Schrödinger equations with step-function coefficients having a finite number of discontinuities. The local and global dispersion and Strichartz inequalities are discussed for certain Schrödinger equations with low regularity coefficients oscillating at infinity.