Stochastic partial differential equations with unbounded and degenerate coefficients

In this article, using DiPerna-Lions theory (DiPerna and Lions, 1989) [1], we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the L¹-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.     

A non-local regularization of first order Hamilton-Jacobi equations

In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C² in space and C¹ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W^1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C^∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L^∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.     

Strong solutions to the Richards equation in the unsaturated zone

We study the unsaturated case of the Richards equation in three space dimensions with Dirichlet boundary data. We first establish an a priori L^∞-estimate. With its help, by means of a fixed point argument we prove global in time existence of a unique weak solution in Sobolev spaces. Finally, we are able to improve the regularity of this weak solution in order to gain a strong one.     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

Variational solutions for partial differential equations driven by a fractional noise

In this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D⊂R^d and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an L²(D)-valued fractional Wiener process W^H with Hurst parameter , whose covariance operator satisfies appropriate integrability conditions, and where γ∈(0,1] denotes the Hölder exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W^H.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

On the periodic Wigner-Poisson-Fokker-Planck system

This paper is concerned with the existence and uniqueness analysis of global classical solutions of a diffusive quantum evolution equation with nonlinear coupling to the Poisson equation. The main technical difficulty in the existence proof is to show that the quantum Fokker-Planck term is a semigroup-generator in a weighted L²-space. The potential term is then a Lipschitz perturbation of it.     

Periodic solutions in unbounded domains for the Boussinesq system

Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn. Our analysis is based on the framework of weak-Lp spaces.     

Non-monotone stochastic generalized porous media equations

By using the Nash inequality and a monotonicity approximation argument, existence and uniqueness of strong solutions are proved for a class of non-monotone stochastic generalized porous media equations. Moreover, we prove for a large class of stochastic PDE that the solutions stay in the smaller L²-space provided the initial value does, so that some recent results in the literature are considerably strengthened.     

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.     

Inhomogeneous Navier–Stokes equations in the half-space,with only bounded density

In this paper, we establish the global existence and uniqueness of solutions to the inhomogeneous Navier–Stokes system in the half-space. The initial density only has to be bounded and close enough to a positive constant, the initial velocity belongs to some critical Besov space, and the L^∞$L^{\infty }$ norm of the inhomogeneity plus the critical norm to the horizontal components of the initial velocity has to be very small compared to the exponential of the norm to the vertical component of the initial velocity. With a little bit more regularity for the initial velocity, those solutions are proved to be unique. In the last section of the paper, our results are partially extended to the bounded domain case.     

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L²(Ω), and hence, the associated first order Cauchy problem is well posed on Lq(Ω) for every q∈[1,∞). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the Lr-norm of the vertical velocity v for any 1<r<∞ is globally bounded and that the L^∞-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace δ(−Δ) for δ>0 would guarantee the global regularity of classical solutions.     

L p solutions of backward stochastic Volterra integral equations

This paper is devoted to the unique solvability of backward stochastic Volterra integral equations (BSVIEs, for short), in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in L ^p (1 < p < 2), which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in L ^p (p > 1) is also considered, which also generalizes the results in the existing literature.     

Existence and scattering theory for Boussinesq type equations with singular data

We study the initial value problem for the generalized Boussinesq equation and prove existence of local and global solutions with singular initial data in weak-Lp spaces. Our class of initial data for global existence is larger than that of Cho and Ozawa (2007) [7]. Long time behavior results are obtained and a scattering theory is proved in that framework. With more structure, we show Sobolev H¹ and Lorentz-type L(p,q) regularity properties for the obtained solutions. The approach employed is unified for all dimensions n?1.     

Relativistic Euler equations for isentropic fluids: Stability of Riemann solutions with large oscillation

We analyze global entropy solutions of the 2 × 2 relativistic Euler equations for isentropic fluids in special relativity and establish the uniqueness of Riemann solutions in the class of entropy solutions in L BV_loc with arbitrarily large oscillation. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions implies their inviscid time-asymptotic stability under arbitrarily large L¹ L BV_loc perturbation of the Riemann initial data, provided that the corresponding solutions are in L and have local bounded total variation that allows the linear growth in time. This approach is also extended to deal with the stability of Riemann solutions containing vacuum in the class of entropy solutions in L with arbitrarily large oscillation.Received: October 21, 2003     

Quasilinear parabolic stochastic partial differential equations: Existence,uniqueness

We present a direct approach to existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone. The proof of uniqueness is very elementary, based on a new method of applying Itô’s formula for the $L^1$-norm. The proof of existence relies on a recent regularity result and is direct in the sense that it does not rely on the stochastic compactness method.     

Nevanlinna-Pick meromorphic interpolation: The degenerate case and minimal norm solutions

The Nevanlinna-Pick interpolation problem is studied in the class Sκ of meromorphic functions f with κ poles inside the unit disk D and with ‖f_‖L^∞(T)?1. In the indeterminate case, the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges. A necessary and sufficient condition for the problem being determinate is given in terms of the Pick matrix of the problem. The result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies Nevanlinna-Pick interpolation conditions and has the minimal possible L^∞-norm on the unit circle T.     

Fourier multipliers and periodic solutions of delay equations in Banach spaces

In this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions. The conditions are obtained in terms of R-boundedness of linear operators determined by the equations and Lp- Fourier multipliers. Periodic mild solutions are also studied and characterized.