Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Given a control region Ω on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Ω. It is known that any initial data in L²(M) can be steered to 0 in an arbitrarily small time T by applying a suitable control g in L2([0,T]×Ω), and, as T tends to 0, the norm of g grows like exp(C/T) times the norm of the data. We investigate how C depends on the geometry of Ω. We prove C?d²/4 where d is the largest distance of a point in M from Ω. When M is a segment of length L controlled at one end, we prove for some . Moreover, this bound implies where is the length of the longest generalized geodesic in M which does not intersect Ω. The control transmutation method used in proving this last result is of a broader interest.     

On the null-controllability of the heat equation in unbounded domains

We make two remarks about the null-controllability of the heat equation with Dirichlet condition in unbounded domains. Firstly, we give a geometric necessary condition (for interior null-controllability in the Euclidean setting) which implies that one cannot go infinitely far away from the control region without tending to the boundary (if any), but also applies when the distance to the control region is bounded. The proof builds on heat kernel estimates. Secondly, we describe a class of null-controllable heat equations on unbounded product domains. Elementary examples include an infinite strip in the plane controlled from one boundary and an infinite rod controlled from an internal infinite rod. The proof combines earlier results on compact manifolds with a new lemma saying that the null-controllability of an abstract control system and its null-controllability cost are not changed by taking its tensor product with a system generated by a non-positive self-adjoint operator.     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.     

Subcritical Kuramoto-Sivashinsky-type equation on a half-line

In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto-Sivashinsky-type equation

(0.1)     

Remarks on polynomial expansions of solutions of the heat equation in two space variables

Results on polynomial expansions of analytic solutions of the heat equation can be used for the discussion of the continuation of analytic solutions. A system of polynomial solutions introduced by Col ton and Wimp [3] is found good for such investigations in the two dimensional case. A Banach scales approach is the base for the results of the present paper     

On the convergence of boundary element methods for initial-Neumann problems for the heat equation

In this paper we study boundary element methods for initial-Neumann problems for the heat equation. Error estimates for some fully discrete methods are established. Numerical examples are presented.

     

On the convergence of difference schemes for a heat conduction equation

Sufficient conditions are obtained for the convergence of difference schemes for the numerical solution of the Cauchy problem for a heat conduction equation in two space variables. The sufficient conditions are derived in a form similar to those for the convergence of a sequence of linear positive operators in the Korovkin theorem. As an application it is shown that difference schemes that are widely used in practice can easily be checked for convergence by these conditions.     

A note on global optimization via the heat diffusion equation

We consider the interesting smoothing method of global optimization recently proposed in Lau and Kwong (J Glob Optim 34:369–398, 2006) . In this method smoothed functions are solutions of an initial-value problem for a heat diffusion equation with external heat source. As shown in Lau and Kwong (J Glob Optim 34:369–398, 2006), the source helps to control global minima of the smoothed functions—they are not shifted during the smoothing. In this note we point out that for certain (families of) objective functions the proposed method unfortunately does not affect the functions, in the sense, that the smoothed functions coincide with the respective objective function. The key point here is that the Laplacian might be too weak in order to smooth out critical points.     

On the determining equations for the nonclassical reductions of the heat and Burgers' equation

The determining equations for the nonclassical reductions of the heat and Burgers' equations are considered. It is shown that both systems belong to a Burgers' equation hierarchy. Each system is written in terms of the same matrix Burgers' equation that is linearized via a matrix Hopf–Cole transformation. In essence, it is shown that both systems can be solved simultaneously. Their respective solutions are then presented in a very compact form.     

Improved entropy decay estimates for the heat equation

Improved entropy decay estimates for the heat equation are obtained by selecting well-parametrized Gaussians. Either by mass centering or by fixing the second moments or the covariance matrix of the solution, relative entropy toward these Gaussians is shown to decay with better constants than classical estimates.     

Some regularity results on the ‘relativistic’ heat equation

We prove some partial regularity results for the entropy solution u of the so-called relativistic heat equation. In particular, under some assumptions on the initial condition u₀, we prove that ut(t) is a Radon measure in RN. Moreover, if u₀ is log-concave inside its support Ω, Ω being a convex set, then we show the solution u(t) is also log-concave in its support Ω(t). This implies its smoothness in Ω(t). In that case we can give a simpler characterization of the notion of entropy solution.     

Modeling the heating of biological tissue based on the hyperbolic heat transfer equation

In modern surgery, a multitude of minimally intrusive operational techniques are used which are based on the point heating of target zones of human tissue via laser or radiofrequency currents. Traditionally, these processes are modeled by the bioheat equation introduced by Pennes, who considers Fourier’s theory of heat conduction. We present an alternative and more realistic model established using the hyperbolic equation of heat transfer. To demonstrate some features and advantages of our proposed method, we apply the results obtained to different types of tissue heating with high energy fluxes, in particular radiofrequency heating and pulsed laser treatment of the cornea to correct refractive errors. We hope that the results from our approach will help with refining surgical interventions in this novel field of medical treatment.     

A note on the complexity and tractability of the heat equation

We wish to solve the heat equation u_t=Δu-qu in I^d×(0,T), where I is the unit interval and T is a maximum time value, subject to homogeneous Dirichlet boundary conditions and to initial conditions u(·,0)=f over I^d. We show that this problem is intractable if f belongs to standard Sobolev spaces, even if we have complete information about q. However, if f and q belong to a reproducing kernel Hilbert space with finite-order weights, we can show that the problem is tractable, and can actually be strongly tractable.     

Recovery of the heat equation from a single boundary measurement

We prove that we can uniquely recover the coefficient of a one-dimensional heat equation from a single boundary measurement and provide a constructive procedure for its recovery. The algorithm is based on the well-known Gelfand–Levitan–Gasymov inverse spectral theory of Sturm–Liouville operators.     

Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation

In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)?−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M)?−k, which generalizes a result of L. Ni (2004, 2006) [20] and [21]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.     

A linear quadratic control problem for the stochastic heat equation driven by Q-Wiener processes

We consider a control problem for the stochastic heat equation with Neumann boundary condition, where controls and noise terms are defined inside the domain as well as on the boundary. The noise terms are given by independent Q-Wiener processes. Under some assumptions, we derive necessary and sufficient optimality conditions stochastic controls have to satisfy. Using these optimality conditions, we establish explicit formulas with the result that stochastic optimal controls are given by feedback controls. This is an important conclusion to ensure that the controls are adapted to a certain filtration. Therefore, the state is an adapted process as well.     

The heat equation to generate planar grids

An algorithm for the generation of quadrilateral grids on planar domains is presented. This algorithm is given by an iterative procedure, where, starting from an initial grid on the domain under consideration, the coordinates of the grid vertices are iteratively adjusted by using a local discrete variational approach. This procedure resembles the explicit difference scheme for a perturbed heat equation, where the perturbation can be dropped for convex domains. Experimental results on benchmark domains are presented, and show an interesting behavior of the proposed method.     

The Li-Yau-Hamilton estimate and the Yang-Mills heat equation on manifolds with boundary

The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold M with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution ∇(t) of the Yang-Mills heat equation in a vector bundle over M. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply that the curvature of ∇(t) does not blow up if the dimension of M is less than 4 or if the initial energy of ∇(t) is sufficiently small.     

On the regularity of the heat equation solution in non-cylindrical domains: Two approaches

In this work, we investigate the behavior of the solution of the Cauchy-Dirichlet problem for a parabolic equation set in a three-dimensional domain with edges. We also give new regularity results for the weak solution of this equation in terms of the regularity of the initial data.     

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-K-invariant $L^1$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-K-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-K-invariance condition and strong $L^{\infty }$ convergence.