In a previous article of Dos Santos Ferreira, Kenig, Salo and Uhlmann, anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n \geq 3. In this article we extend this result to the case of unbounded potentials, namely those in Ln/2. In the process, we derive Lp Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison-Kenig and Kenig-Ruiz-Sogge.
We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton-Jacobi-Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the Value Function of the controlled equation and that the feedback law is verified.
We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method provided by E.E. Levi.
We prove in this short report the existence of a fundamental solution (F.S.) for the Cauchy initial boundary problem on the whole space for the parabolic differential equation having at origin the point of non-integrable unbounded discontinuity for coefficient before a first order derivative. We give also the non-asymptotic rapidly decreasing at infinity estimate for these function. We extend the classical parametrix method offered by E.E.Levi.
This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $Ω\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-Δ+1)β(u) = g \quad \mbox{in}\ Ω\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $β$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $β$ even tho…
We study the Cauchy problem for Fokker--Planck--Kolmogorov equations with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are indicated.
We construct solutions to p-Laplace type equations in unbounded Lipschitz domains in the plane with prescribed boundary data in appropriate fractional Sobolev spaces. Our approach builds on a Cauchy integral representation formula for solutions.
We consider the Cauchy problem for one-dimensional (1D) barotropic compressible Navier-Stokes equations with density-dependent viscosity and large external force. Under a general assumption on the density-dependent viscosity, we prove that the Cauchy problem admits a unique global strong (classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently. As a consequence, we obtain the large time behavior of the solution without external forces.
We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz'ya proposed an alternating iterative method for solving Cauchy problems associated with elliptic,self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We cons…
The purpose of this paper is to obtain an upper bound for the fundamental solution for parabolic Cauchy problem u'=Au, where A is a second order elliptic partial differential operator with unbounded coefficients such that its potential and the potential of the formal adjoint of the operator A are bounded from below.
We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in $x$ with their $x$-derivatives and $t$-derivative of order ${O}(t^{-δ}),δ\in [0,1),$ and ${O}(t^{-1})$ respectively. This type of singular behavior allows coefficients to be either oscillatory or logarithmically bounded at $t=0$. We use the Planck function associated with the …
We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions.
We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence result of a weak solution. In this case we never impose strong assumptions such as compactness of the support of the initial data. This means that we never rely on the finite propagation speed property of the solution, and we try to deal with an essential unbounded coefficient case.
The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{t\ge0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t)_{t\ge0}$ extends uniquely as a continuous semi-group over $L^p(\mathbb{R}^n)$ whenever $1\le p<\infty$, and $u(t):=S_tu_0$ is actually an entropy solution to the Cauchy problem. When $p\le q\le \infty$ and $t>0$, $S_t$ actually maps $L^p(\mathbb{R}^n)$ into $L^q(\mathbb{R}^n)$. These results are based upon new dispers…
We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast …
The present paper is concerned with Schrödinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (resp. magnetic) potential can grow subquadratically (resp. sublinearly) at spatial infinity. We prove sharp (local-in-time) Strichartz estimates, outside a large compact ball centered at origin, for any admissible pair including the endpoint. Under the nontrapping condition on the Hamilton flow generated by the kinetic energy, global-in-space estimates are also studied. F…
We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the decay of the solution as time increases, in relation with the nonlinearity. The main ingredient is our recent theory of divergence-free positive symmetric tensor. We apply in particular the so-called compensated integrability to a tensor which generalizes the one that L. Tartar used in one space dimension. It allows us to establish a Strichartz-like inequality…
We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan-Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also…
We study the Cauchy problem for the semilinear nonautonomous parabolic equation $u_t=\mathcal{A}(t)u+ψ(t,u)$ in $[s,τ]\times {{\mathbb R}^d}$, $τ> s $, in the spaces $C_b([s, τ]\times{{\mathbb R}^d})$ and in $L^p((s, τ)\times{{\mathbb R}^d}, ν)$. Here $ν$ is a Borel measure defined via a tight evolution system of measures for the evolution operator $G(t,s)$ associated to the family of time depending second order uniformly elliptic operators $\mathcal{A}(t)$. Sufficient conditions for existence in the large and stability of the null solution are also given in both $C_b$ and $L^p$ contexts. The …
The Cauchy problem for a multidimensional linear transport equation with unbounded drift is investigated. Provided the drift is Holder continuous , existence, uniqueness and strong stability of solutions are obtained. The proofs are based on a careful analysis of the associated stochastic flow of characteristics and techniques of stochastic analysis.
On the Cauchy problem for the Langevin-type fractional equation
National University of Uzbekistan（乌兹别克斯坦国立大学）；V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）；Karshi State University（卡尔希国立大学）
We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | λ_i(u)| \geq c > 0 {for all} i\in \{1,...,n\}, \|g(x,\cdot)\|_{\mathbf{C}^2}\leq \tilde M(x) \in L1, {array}. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the $\mathbf{L}^1$ norm of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{BV(\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total varia…
We study singular perturbations of a class of two-scale stochastic control systems with unbounded data. The assumptions are designed to cover some relaxation problems for deep neural networks. We construct effective Hamiltonian and initial data and prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of probability, viscosity solutions and homogenization.
In this paper we study the Cauchy problem for semi-linear de Sitter models with power non-linearity. The model of interest is \[ ϕ_{tt} - e^{-2t} Δϕ+ nϕ_t+m^2ϕ=|ϕ|^p,\quad (ϕ(0,x),ϕ_t(0,x))=(f(x),g(x)),\] where $m^2$ is a non-negative constant. We study the global (in time) existence of small data solutions. In particular, we show the interplay between the power $p$, admissible data spaces and admissible spaces of solutions (in weak sense, in sense of energy solutions or in classical sense).
In order to describe behavior of various liquid-like materials at high pressures, incompressible fluid models with pressure dependent viscosity seem to be a suitable choice. In the context of implicit constitutive relations involving the Cauchy stress and the velocity gradient these models are consistent with standard procedures of continuum mechanics. Understanding mathematical properties of governing equations is connected with various types of idealization, some of them lead to studies in unbounded domains. In this paper, we first bring up several characteristic features concerning fluids w…
We consider elliptic equations of Schrödinger type with a right-hand side fixed and with the linear part of order zero given by a potential V . The main goal is to study the optimization problem for an integral cost depending on the solution uV , when V varies in a suitable class of admissible potentials. These problems can be seen as the natural extension of shape optimization problems to the framework of potentials. The main result is an existence theorem for optimal potentials, and the main difficulty is to work in the whole Euclidean space Rd, which implies a lack of compactness in several…
In this article, we prove the global (in time) existence of small data solutions from energy spaces basing on $L^q$ spaces, with $q \in (1,\infty)$, to the Cauchy problems for a weakly coupled system of semi-linear visco-elastic damped $σ$-evolution models. Here we consider different power nonlinearities and different $σ$ values in the comparison between two single equations. To do this, we use $(L^m \cap L^q)- L^q$ and $L^q- L^q$ estimates, i.e., by mixing additional $L^m$ regularity for the data on the basis of $L^q- L^q$ estimates for solutions, with $m \in [1,q)$, to the corresponding line…
The purpose of this article is to extend the uniqueness results for the two dimensional Calderón problem to unbounded potentials on general geometric settings. We prove that the Cauchy data sets for Schrödinger equations uniquely determines potentials in $L^{p}$ for $p> 4/3$. In doing so, we first recover singularities of the potential, from which point a $L^2$-based method of stationary phase can be applied. Both steps are done via constructions of complex geometric optic solutions and Carleman estimates.
We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.
Stability in recovering the order of a time-fractional evolution equation
V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）；School of Engineering, Central Asian University（中亚大学工程学院）；Graduate School of Mathematical Sciences, The University of Tokyo（东京大学数理科学研究科）；Department of Mathematics, Faculty of Arts and Sciences, Zonguldak Bülent Ecevit University（宗古尔达克比伦特杰奇维特大学文理学院数学系）
This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible (in particular, the $C^{0}$-formulation of…
We study inverse boundary problems for magnetic Schrödinger operators on a compact Riemannian manifold with boundary of dimension $\ge 3$. In the first part of the paper we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Cauchy data on the boundary of the manifold for the magnetic Schrödinger operator with $L^\infty$ magnetic and electric potentials, determines the magnetic field and electric potential uniquely. In the …
We consider the wave equation on Reissner-Nordström-de Sitter and more generally Kerr-Newman-de Sitter black hole spacetimes with $Λ>0$. The strength of the blue-shift instability associated to the Cauchy horizon of these spacetimes has been the subject of much discussion, since-in contrast to the $Λ=0$ case-the competition with the decay associated to the region between the event and cosmological horizons is delicate. Of particular interest is the question as to whether generic, admissible initial data posed on a Cauchy surface lead to solutions whose local energy blows up at the Cauchy horiz…
This is the second and last paper of a two-part series in which we prove the $C^2$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. In the first paper (arXiv:1702.05715), we showed that the maximal globally hyperbolic future development of an admissible asymptotically flat Cauchy initial data set is $C^2$-future-inextendible provided that an $L^2$-averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show …
