The Construction of an Evolution System in the Hyperbolic Case and Applications

We study the Cauchy problem for abstract linear and quasi–linear non–autonomous evolution equations of hyperbolic type using semigroup theory. Under weak differentiability assumptions on the time regularity of the coefficients we prove well–posedness and regularity of a solution. The abstract results are illustrated by their application to a series of equations of mathematical physics.     

Global well‐posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

In this paper, we discuss with the global well‐posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well‐posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model, that is, F(θ) = θe₂. Copyright © 2017 John Wiley & Sons, Ltd.     

Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains

We consider an elliptic system in divergence form with measurable coefficients in a nonsmooth bounded domain to find a minimal regularity requirement on the coefficients and a lower level of geometric assumption on the boundary of the domain for a global W ^1,p , 1 < p < ∞, regularity. It is proved that such a W ^1,p regularity is still available under the assumption that the coefficients are merely measurable in one variable and have small BMO semi-norms in the other variables while the domain can be locally approximated by a hyperplane, a so called δ-Reifenberg domain, which is beyond the Lipschitz category. This regularity easily extends to a certain Orlicz-Sobolev space.     

Initial-Boundary Value Problem for a Class of Linear Relaxation Systems in Arbitrary Space Dimensions

In this paper, we consider the characteristic initial-boundary value problem (IBVP) for the multi-dimensional Jin-Xin relaxation model in a half-space with arbitrary space dimension n?2. As in the one-dimensional case (n=1, see (J. Differential Equations, 167 (2000), 388-437), our main interest is on the precise structural stability conditions on the relaxation system, particularly the formulation of boundary conditions, such that the relaxation IBVP is stiffly well posed, that is, uniformly well posed independent of the relaxation parameter ε>0, and the solution of the relaxation IBVP converges, as ε→0, to that of the corresponding limiting equilibrium system, except for a sharp transition layer near the boundary. Our main result can be roughly stated as Stiff Kreiss Condition=Uniform Kreiss Condition for the relaxation IBVP we consider in this paper, which is in sharp contrast to the one-dimensional case (Z. Xin and W.-Q. Xu, J. Differential Equations, 167 (2000), 388-437). More precisely, we show that the Uniform Kreiss Condition (which is necessary and sufficient for the well posedness of the relaxation IBVP for each fixed ε), together with the subcharacteristic condition (which is necessary and sufficient for the stiff well posedness of the corresponding Cauchy problem), also guarantees the stiff well posedness of our relaxation IBVP and the asymptotic convergence to the corresponding equilibrium system in the limit of small relaxation rate. Optimal convergence rates are obtained and various boundary layer behaviors are also rigorously justified.     

Fourier series and integral equation method for the exterior Stokes problem

The exterior Stokes problem between two parallel planes that are separated by a prismatic cylinder is extended to the interior of the prism by requiring the continuity of the velocity across the lateral faces. The well‐posedness of the exterior–interior problem is proved in suitable weighted Sobolev spaces. The solution is represented by Fourier series in the z‐variable. The Fourier coefficients, solutions of auxiliary two‐dimensional exterior–interior problems, are analyzed by viewing them as boundary integral equations of potential theory and global regularity of the densities, is established in weighted Sobolev spaces of traces. A boundary element method, with suitably refined mesh size, is implemented for the numerical treatment of the Fourier coefficients. This provides optimal convergent semi‐ and fully‐discrete spectral methods of Fourier–Galerkin type. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: “smoothness”, “oscillations”, “degeneration” and “stabilization”. Actually, we prove the Gevrey and C^∞ well‐posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-posedness and regularity of boundary feedback parabolic systems

A general second order parabolic equation is considered with both Dirichlet or mixed (in particular, Neumann) input function acting on the boundary S of the bounded spatial domain Ω. The distinctive new feature is that the input function is demanded to be expressed in feedback form, i.e. as a linear operator (of finite dimensional range) of the solution, continuous from $\text{H}{}^{\mathrm{s}}\text{(Ω)}$ into L_p(S), for some non-negative real s and for p ? 1. Well posedness and regularity results of the resulting closed loop system are established in appropriate functions spaces. The results are illustrated by examples of physical interest.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Loss of regularity for p-evolution type models

The goal of the paper is to study the loss of regularity for special p-evolution type models with bounded coefficients in the principal part. The obtained loss of regularity is related in an optimal way to some unboundedness conditions for the derivatives of coefficients up to the second-order with respect to t.     

Regularity criteria for almost every function in Sobolev spaces

In this paper we determine the multifractal nature of almost every function (in the prevalence setting) in a given Sobolev or Besov space according to different regularity exponents. These regularity criteria are based on local Lp regularity or on wavelet coefficients and give a precise information on pointwise behavior.     

Initial–boundary value problems for continuity equations with BV coefficients

We establish well-posedness of initial–boundary value problems for continuity equations with BV (bounded total variation) coefficients. We do not prescribe any condition on the orientation of the coefficients at the boundary of the domain. We also discuss some examples showing that, regardless of the orientation of the coefficients at the boundary, uniqueness may be violated as soon as the BV regularity deteriorates at the boundary.     

Mathematical analysis for an epidemic model with spatial and age structure

A system of parabolic–hyperbolic equations with a non-local boundary condition, arising in mathematical theory of epidemics, is analyzed. For such a system, well–posedness as well as Sobolev regularity with respect to the space variables is proved. Asymptotic behavior of the solutions is also investigated.     

On the Cauchy problem for second order strictly hyperbolic equations with non–regular coefficients

In this paper we shall consider some necessary and sufficient conditions for well–posedness of second order hyperbolic equations with non–regular coefficients with respect to time. We will derive some optimal regularities for well–posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory.     

Ideal density‐dependent incompressible viscoelastic flow in the critical Besov spaces

In this paper, we consider the well‐posedness issue for the density‐dependent incompressible viscoelastic fluids of the Oldroyd model for the ideal case in space dimension greater than 2. We obtain the local well‐posedness of this model under the assumption that the initial density is bounded away from zero in the critical Besov spaces by means of the Littlewood‐Paley theory and Bony's paradifferential calculus. In particular, we obtain a Beale‐Kato‐Majida–type regularity criterion.     

Estimation adaptative de la densité avec données échantillonnées

In continuous time, rates of convergence for nonparametric density estimators depend on the nature of sample paths: roughly speaking, the more ‘irregular’ the paths are, the better the rates are. In this framework, we give the pointwise rate of convergence of the kernel density estimator in the case of sampled observations. Behaviour of the estimator depends on two coefficients r₀, γ₀ respectively linked with regularity of density and regularity of sample paths. We propose an adaptive estimator relatively to γ₀ as well as a doubly adaptive estimator (with respect to r₀ and γ₀). It is shown that the rate of convergence obtained in the case of known r₀, γ₀ is achieved by such adaptive estimators. To cite this article: D. Blanke, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Decay and regularity in the inverse scattering problem

The regularity and decay properties for the potential q(x) in the Schrödinger equation ?ψ″ + qψ = k²ψ on the line are characterized in terms of the decay and regularity of the reflection coefficients R_± and their Fourier transforms.     

Differential equations with singular fields

This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow (bounded Jacobian) is considered. The main result provides existence under the condition that the vector field belongs to BV in dimension 2 and SBV in higher dimensions.     

Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients

We prove existence of strongly continuous evolution systems in L² for Schrödinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral space variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity result, which enables us to precisely determine the common domain of the generators. The construction of a solution with low regularity in the coefficients is the basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data.     

On the low regularity of the Benney-Lin equation

We consider the low regularity of the Benney-Lin equation ut+uux+uxxx+β(uxx+uxxxx)+ηuxxxxx=0. We established the global well posedness for the initial value problem of Benney-Lin equation in the Sobolev spaces Hs(R) for 0?s>−2, improving the well-posedness result of Biagioni and Linares [H.A. Biaginoi, F. Linares, On the Benney-Lin and Kawahara equation, J. Math. Anal. Appl. 211 (1997) 131-152]. For s<−2 we also prove some ill-posedness issues.     

An inverse diffusion problem with nonlocal boundary conditions

This article considers the inverse problem of identification of a time‐dependent thermal diffusivity together with the temperature in an one‐dimensional heat equation with nonlocal boundary and integral overdetermination conditions when a heat exchange takes place across boundary of the material. The well‐posedness of the problem is studied under some regularity, and consistency conditions on the data of the problem together with the nonnegativity condition on the Fourier coefficients of the initial data and source term. The inverse problem is also studied numerically by using the Crank–Nicolson finite difference scheme combined with predictor‐corrector technique. The numerical examples are presented and discussed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 564–590, 2016