In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.
In this work we deal with the selection problem of flows of an irregular vector field. We first summarize an example from \cite{CCS} of a vector field $b$ and a smooth approximation $b_ε$ for which the sequence $X^ε$ of flows of $b_ε$ has subsequences converging to different flows of the limit vector field $b$. Furthermore, we give some heuristic ideas on the selection of a subclass of flows in our specific case.
Weakly harmonic maps from a domain $Ω$ (the upper half-space $\Rd$ or a bounded $C^{1,α}$ domain, $α\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes $L^{\infty}(\partialΩ)$ or $BMO(\partialΩ)$, we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, $C^{\infty}_{loc}$. Moreover, we show that boundary data can be chosen large in the underlying topologies if $Ω$ is smooth and bounded by perturbing strictly stable smooth harmonic…
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.
We present a general approach to obtain a weak Harnack inequality for rough hypoellipitic equations, e.g. kinetic equations. The proof is constructive and does not study the commutator structure but rather compares the rough solution with a smooth problem for which the estimates are assumed.
\rm We obtain the global smooth effects for the solutions of the linear Schrödinger equation in anisotropic Lebesgue spaces. Applying these estimates, we study the Cauchy problem for the generalized elliptical and non-elliptical derivative nonlinear Schrödinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces $M^{3/2}_{2,1}(\mathbb{R}^n)$. Noticing that $H^{\tilde{s}} \subset M^s_{2,1}$ $(\tilde{s}-s>n/2)$ is an optimal inclusion, we have shown the global well posedness of DNLS with a class of very rough data.
In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous Dirichlet boundary condition is imposed; we assume boundedness of the embedding $i_{1}:H^{1}(D)\to L^{2}(D)$ when the Neumann boundary condition is imposed; we assume boundedness of the embeddings $i_{1}$ and of $i_{2}:H^{1}(D)\to L^{2}(\partial D)$ when the Robin boundary condition is imposed, and, if, in addition, $i_{1}$ and $i_{2}$ are compact, then the boun…
In this paper, we consider viscoelastic flows in a rough domain (with typical roughness patterns of size $ε$ < 1). We present and rigorously justify an asymptotic expansion with respect to $ε$, at any order, based upon the definition of elementary problems: Oldroyd-type problems at the global scale defined on a smoothened domain and boundary-layer corrector problems. The resulting analysis guarantees optimality with respect to the truncation error.
In this paper we study boundary value problems for higher order elliptic differential operators in divergence form. We consider the two closely related topics of inhomogeneous problems and problems with boundary data in fractional smoothness spaces. We establish $L^\infty$ perturbative results concerning well posedness of inhomogeneous problems with boundary data in fractional smoothness spaces. Combined with earlier known results, this allows us to establish new well posedness results for second order operators whose coefficients are close to being real and t-independent and for fourth-order …
We give an example of a bounded divergence free autonomous vector field in $\mathbb R^3$ (and of a nonautonomous bounded divergence free vector field in $\mathbb R^2$) and of a bounded initial data for which the Cauchy problem for the corresponding transport equation has $2$ distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.
In the class of Sobolev vector fields in $\mathbb{R}^n$ of bounded divergence, for which the theory of DiPerna and Lions provides a well defined notion of flow, we characterize the vector fields whose flow commute in terms of the Lie bracket and of a regularity condition on the flows themselves. This extends a classical result of Frobenius in the smooth setting.
It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time (Ph. Serfati C.R. Acad. Sci. Série I 320, 175-180 (1995); A. Shnirelman arXiv:1205.5837 (2012)). Here an elementary derivation is given, based on Cauchy's form of the Euler equations in Lagrangian coordinates. This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the C^{1,γ} Hölder norms of t…
For a bounded domain $Ω\subset\mathbb{R}^m, m\geq 2,$ of class $C^0$, the properties are studied of fields of `good directions', that is the directions with respect to which $\partialΩ$ can be locally represented as the graph of a continuous function. For any such domain there is a canonical smooth field of good directions defined in a suitable neighbourhood of $\partialΩ$, in terms of which a corresponding flow can be defined. Using this flow it is shown that $Ω$ can be approximated from the inside and the outside by diffeomorphic domains of class $C^\infty$. Whether or not the image of a gen…
A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the loss of smoothness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,α}$. Moreover, we show by means of this shear flow exa…
We prove an optimal dispersive $L^{\infty}$ decay estimate for a three dimensional wave equation perturbed with a large non smooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator.
New classes of non-smooth bounded domains D, for which the embedding operator from H^1(D) into L^2(D) is compact are introduced. Examples are given and applications to scattering by rough obstacles are mentioned.
We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Holder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, unique…
We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreover, every rough, maximal, asymptotically Euclidean solution can be approximated in an appropriate topology by smooth solutions. These results have application in an existence theorem for rough solutions of the Einstein evolution equations.
We consider the classical limit of the quantum evolution, with some rough potential, of wave packets concentrated near singular trajectories of the underlying dynamics. We prove that under appropriate conditions, even in the case of BV vector fields, the correct classical limit can be selected.
