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.     

Local a priori estimates in Lp for first order linear operators with nonsmooth coefficients

We prove local a priori estimates inL ^p , 1<p<∞, for first-order linear operators that satisfy the Nirenberg-Treves condition (p) and whose coefficients have Lipschitz continuous derivatives of order one. When the number of variables is two, only Lipschitz continuity of the coefficients is assumed. This extends toL ^p spaces estimates that were previously known forp=2. Examples show that the regularity required from the coefficients is essentially minimal. Research partially supported by CNPq.     

Sharp estimates for a class of hyperbolic pseudo-differential equations

In this paper we consider the Cauchy problem for a class of hyperbolic pseudodifferential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L ^p ? L^p, Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R¹⁺ⁿ with n ≤ 4 (total dimension ≤ 5), we give a complete list of L ^p ? L^p properties. In particular, this contains the very important case R¹⁺³.     

An abstract transmission problem in a thin layer, I: Sharp estimates

We study an elliptic transmission problem in Banach spaces. The problem is considered on the juxtaposition of two intervals, one of which of small length δ, and models physical phenomena in media constituted by two parts with different physical characteristics. We obtain results of existence, uniqueness, maximal regularity and optimal dependence on the parameter δ for Lp solutions of the problem. The main tools of our approach are impedance and admittance operators (i.e. Dirichlet-to-Neumann and Neumann-to-Dirichlet operators) and H^∞ functional calculus for sectorial operators in Banach spaces.     

A PDE approach to regularity of solutions to finite horizon optimal switching problems

We study optimal 2-switching and n-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the n-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C^1,1-regular away for the free boundaries of the action sets.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

Finite element convergence for the Joule heating problem with mixed boundary conditions

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes.     

Space-time estimates for the wave equations with singular potentials

We study the regularity of the solutions for the wave equations with potentials that are time-dependent and singular. The size of the potentials is exactly a function of the spatial dimension rather than being small enough in the known results. Based on a weighted L² estimate for the solutions, we prove the local regularity and the Strichartz estimates. The solvability of the equation is also studied.     

Optimal algorithms for global optimization in case of unknown Lipschitz constant

We consider the global optimization problem for d-variate Lipschitz functions which, in a certain sense, do not increase too slowly in a neighborhood of the global minimizer(s). On these functions, we apply optimization algorithms which use only function values. We propose two adaptive deterministic methods. The first one applies in a situation when the Lipschitz constant L is known. The second one applies if L is unknown. We show that for an optimal method, adaptiveness is necessary and that randomization (Monte Carlo) yields no further advantage. Both algorithms presented have the optimal rate of convergence.     

A Note on Time-Decay Estimates for the Compressible Navier–Stokes Equations

In the recent work, we have developed a decay framework in general L~p critical spaces and established optimal time-decay estimates for barotropic compressible Navier–Stokes equations. Those decay rates of L~q-L~r type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions.     

Optimal decay rates for the compressible fluid models of Korteweg type

We consider the compressible Navier-Stokes-Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the optimal L² and Lp, p?2 decay rates for the classical solutions and their spatial derivatives. In particular, the optimal L² decay rate of the second-order spatial derivatives is obtained. The proof is based on the detailed study of the linear decay estimates and nonlinear energy estimates.     

Weak solutions for the exterior Stokes problem in weighted Sobolev spaces

We extend here some existence and uniqueness results for the exterior Stokes problem in weighted Sobolev spaces. We also study the regularity of the solutions ( u ,π) and prove optimal a priori estimates for the solutions with ∇² u ,∇π∈L^p. The influence of some compatibility conditions on the behaviour at infinity of the solution is finally studied and leads to new asymptotic expansions. Copyright © 2000 John Wiley & Sons, Ltd.     

Regularity of variational solutions to linear boundary value problems in Lipschitz domains

In a bounded Lipschitz domain in ?ⁿ, we consider a second-order strongly elliptic system with symmetric principal part written in divergent form. We study the Neumann boundary value problem in a generalized variational (or weak) setting using the Lebesgue spaces H _p ^σ (Ω) for solutions, where p can differ from 2 and σ can differ from 1. Using the tools of interpolation theory, we generalize the known theorem on the regularity of solutions, in which p = 2 and {σ ? 1} < 1/2, and the corresponding theorem on the unique solvability of the problem (Savaré, 1998) to p close to 2. We compare this approach with the nonvariational approach accepted in numerous papers of the modern theory of boundary value problems in Lipschitz domains. We discuss the regularity of eigenfunctions of the Dirichlet, Neumann, and Poincaré-Steklov spectral problems.     

Estimations intérieures avec régularité optimale pour un modèle de plaques en élasticité linéaireInterior estimates with optimal regularity for a plate model in linear elasticity

We are interested in the 3d–2d passage for an asymptotically thin plate in linear elasticity. The classical approach by asymptotic expansion gives an error estimate on the displacements in H¹, assuming the volumic forces at least of regularity L² (and more for certain components). In our work we apply the regularity theory for solutions of elliptic equations. This approach gives, for a new model of Kirchhoff–Love of higher order, an error estimate in H² assuming volumic forces only in L², which is optimal. We also get some interior error estimates in W^k,p, C^k,α. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 207–212.     

Existence of minimizers of the Mumford-Shah functional with singular operators and unbounded data

We consider the regularization of linear inverse problems by means of the minimization of a functional formed by a term of discrepancy to data and a Mumford-Shah functional term. The discrepancy term penalizes the L ² distance between a datum and a version of the unknown function which is filtered by means of a non-invertible linear operator. Depending on the type of the involved operator, the resulting variational problem has had several applications: image deblurring, or inverse source problems in the case of compact operators, and image inpainting in the case of suitable local operators, as well as the modeling of propagation of fracture. We present counterexamples showing that, despite this regularization, the problem is actually in general ill-posed. We provide, however, existence results of minimizers in a reasonable class of smooth functions out of piecewise Lipschitz discontinuity sets in two dimensions. The compactness arguments we developed to derive the existence results stem from geometrical and regularity properties of domains, interpolation inequalities, and classical compactness arguments in Sobolev spaces.     

Global well-posedness for the one-phase Muskat problem

The free boundary problem for a two-dimensional fluid permeating a porous medium is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L_t^{\infty }{L}_x^2$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.     

A Two-Phase Parabolic Free Boundary Problem with Coefficients Below the Lipschitz Threshold

We study the regularity of a parabolic free boundary problem of two-phase type with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the optimal ${C_x^{1,1}\cap C_t^{0,1}}$ -regularity of the solution and that the free boundary is, near the so-called branching points, the union of two graphs that are Lipschitz in time and C ¹ in space. In our case, the same monotonicity formula does not apply in the same way. Instead we use scaling arguments similar to the ones used for the elliptic case in Edquist et al. (Ann Inst Henri Poincareé, Anal Non Linéaire 26(6):2359?C2372, 2009) to prove the optimal regularity. However, whenever the spatial gradient does not vanish on the free boundary, we are in the parabolic setting faced with some extra difficulties, that forces us to strain our assumptions slightly.     

Approximation des branches de solutions d’un problème de bifurcation semi-linéaire

This Note deals with the approximation, by a finite element method, including numerical integration, of solution branches for a semilinear problem. Because of both mixed boundary conditions and geometrical properties of the domain, some of the solutions do not belong to H². So, classical results for convergence lead to poor estimates. We show how to improve such estimates with the use of weighted Sobolev spaces together with a mesh “a priori adapted” to the singularity. For the H¹ or L²-norms, we achieve optimal results, in the sense that they are as good as for H²-regular solutions.     

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

In this paper, we study the local behavior of the solutions to the three-dimensional magnetohydrodynamic equations. we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions. It is shown that, in some neighborhood of (x₀,t₀), the gradients of the velocity field u and the magnetic field B are locally uniformly bounded in L^∞ norm as long as that either the scaled local L²-norm of the gradient or the scaled local total energy of the velocity field is small, and the scaled local total energy of the magnetic field is uniformly bounded. These estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory. As an immediately corollary we can derive an estimates of Hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid. Various partial regularity results are obtained as consequences of our blow-up estimates.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.