This article is dedicated to insensitization issues of a quadratic functional involving the solution of the linear heat equation with respect to domains variations. This work can be seen as a continuation of [P. Lissy, Y. Privat, and Y. Simporé. Insensitizing control for linear and semi-linear heat equations with partially unknown domain. ESAIM Control Optim. Calc. Var., 25:Art. 50, 21, 2019], insofar as we generalize several of the results it contains and investigate new related properties. In our framework, we consider boundary variations of the spatial domain on which the solution of the PDE is defined at each time, and investigate three main issues: (i) approximate insensitization, (ii) approximate insensitization combined with an exact insensitization for a finite-dimensional subspace, and (iii) exact insensitization. We provide positive answers to questions (i) and (ii) and partial results to question (iii).
We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of R^N (N $\in$ N *), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of…
We consider the control problem for the generalized heat equation for a Schroedinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts and the corresponding control cost does not exceed the one on the whole domain. As an application, we obtain null-controllability results for the heat equation on half-spaces, orthants, and sectors of angle $π/2$. As a byproduct, we also obtain explicit control cost bounds for the heat equation on certain triangles and corresponding prisms…
In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control supported on an arbitrary subboundary. The proof of the main result combines a new boundary Carleman estimate and some regularity estimates for the adjoint system, with an explicit dependence with respect to the final time. This technique allows us to overcome a new difficulty that arises when absorbing a normal derivative term.
We survey recent results on the control problem for the heat equation on unbounded and large bounded domains. First we formulate new uncertainty relations, respectively spectral inequalities. Then we present an abstract control cost estimate which improves upon earlier results. It is particularly interesting when combined with the earlier mentioned spectral inequalities since it yields sharp control cost bounds in several asymptotic regimes. We also show that control problems on unbounded domains can be approximated by corresponding problems on a sequence of bounded domains forming an exhausti…
We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $\varepsilon \to 0$, the Froude number proportional to $\sqrt{\varepsilon}$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $\varepsilon \to 0$. The limit velocity field is solenoidal and satisfies the incompressible Oberbeck-Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we apply the spectral analysis of the associated wave propa…
We consider here a generalized heat equation $\partial_t ρ=\frac{d}{dx}\frac{d}{dW}ρ$, where $W$ is a finite measure on the one dimensional torus, and $\frac{d}{dW}$ is the Radon-Nikodym derivative with respect to $W$. Such equation has appeared in different contexts, being related to physical systems and representing a large class of classical and non-classical parabolic equations. As a natural assumption on $W$, we require that the Lebesgue measure is absolutely continuous with respect to $W$. The main result here presented consists in proving, for a suitable topology, a continuous dependenc…
Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $L^p$. Derivation of the coefficients is based on solving certain optimization problems with respect to a vector parameter inside of an integral over the unit sphere.
Let $q\geq 1+\frac{2}{N}$. We prove that any positive solution of (E) $\prt_t u-\xD u+u^q=0$ in $\mathbb{R}^N\times(0,\infty)$ admits an initial trace which is a nonnegative Borel measure, outer regular with respect to the fine topology associated to the Bessel capacity $C_{\frac{2}{q},q'}$ in $\BBR^N$ ($q'=q/q-1)$) and absolutely continuous with respect to this capacity. If $ν$ is a nonnegative Borel measure in $\BBR^N$ with the above properties we construct a positive solution $u$ of (E) with initial trace $\gn$ and we prove that this solution is the unique $\gs$-moderate solution of (E) wit…
It is well known that both the heat equation with Dirichlet or Neumann boundary conditions are null controlable as soon as the control acts in a non trivial domain (i.e. a set of positive measure, see [10, 11, 12, 1, 6]. In this article, we show that for any couple of initial data (u0, v0) we can achieve the null control for both equations (Dirichlet and Neumann boundary conditions respectively) simultaneously with the same control function for both equations.
We consider the initial boundary value problem of non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random perturbation. The space boundary is Lipschitz and we impose non-zero cylinder condition. We prove a regularity result after finding suitable spaces for the solution and the pre-assigned datum in the problem. The tools from potential theory, harmonic analysis and probability are used. Some Lemmas are as important as the main Theorem.
We consider a nonlinear heat equation with a gradient term. We construct a blow-up solution for this equation with a prescribed blow-up profile. For that, we translate the question in selfsimilar variables and reduce the problem to a finite dimensional one. We then solve the finite dimensional problem using index theory. The interpretation of the finite dimensional parameters allows us to derive the stability of the constructed solution with respect to initial data.
We consider the heat equation defined by a generalized measure theoretic Laplacian on $[0,1]$. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure $μ$, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding…
We introduce the transportation-annihilation distance $W_p^\sharp$ between subprobabilities and derive contraction estimates with respect to this distance for the heat flow with homogeneous Dirichlet boundary conditions on an open set in a metric measure space. We also deduce the Bochner inequality for the Dirichlet Laplacian as well as gradient estimates for the associated Dirichlet heat flow. For the Dirichlet heat flow, moreover, we establish a gradient flow interpretation within a suitable space of charged probabilities. In order to prove this, we will work with the \emph{doubling} of the …
Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hörmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$ \mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t}% $$ where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}…
In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any $H^1$ initial data. The initial density is required to be nonnegative, which is not necessary to be uniformly away from vacuum. This not only generalizes the well-known result of Kazhikhov--Shelukhin (Kazhikhov, A.~V.; Shelukhin, V.~V.: \emph{Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas}, J.…
We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The upper bound from our previous work [KR22] is affected by a new correction term measuring the distance to the origin. The main result is then applied to anti-trees with unbounded vertex degree, yielding Gaussian upper bounds for this class of graphs for the first time. In order to prove this, we show that isoperimetric estimates with respect to intrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-…
We discuss reachable states for the Hermite heat equation on a segment with boundary $L^2$-controls. The Hermite heat equation corresponds to the heat equation to which a quadratic potential is added. We will discuss two situations: when one endpoint of the segment is the origin and when the segment is symmetric with respect to the origin. One of the main results is that reachable states extend to functions in a Bergman space on a square one diagonal of which is the segment under consideration, and that functions holomorphic in a neighborhood of this square are reachable.
Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on bd(D), and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in $L^2(D)$ satisfies a strong Hardy inequality with weight $r^2$, (ii) the initial temperature distribution, and the specific heat of D are given by $r^{-a}$ and $r^{-b}$ respectively, where $r$ is the distance to the boundary, and 1<a<2, 1<b<2.
The classical heat equation is incompatible with relativity, since the strong maximum principle allows for disturbances to propagate instantaneously. Some authors have proposed limiting the propagation speed by adding a linear hyperbolic correction term, but then even a weak maximum principle fails to hold. We study a more recently introduced relativistic heat equation, which replaces the Laplace operator by a quasilinear elliptic operator, and show that strong and weak maximum principles hold for stationary and time-varying solutions, respectively, as well as for sub- and supersolutions. More…
We examine the Navier-Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in $R^3$. The considered domain is a cylinder with $x_3$-axis. The aim of this paper is to show long time estimates without smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of $L_2$ norms of derivatives with respect to $x_3$ of the initial velocity, the initial temperature and the external force.
For a heat equation with Robin's boundary conditions which depends on a parameter $α>0$, we prove that its unique weak solution $ρ^α$ converges, when $α$ goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann's boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of $ρ^α$ obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.
