Noncylindrical Domains Research Articles

Finite approximate controllability for semilinear heat equations in noncylindrical domains

We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.

Exponential decay for Kirchhoff wave equation with nonlocal condition in a noncylindrical domain

In this paper, we prove the exponential decay as time goes to infinity of regular solutions of the problem for the Kirchhoff wave equation with nonlocal condition and weak damping u t t − M ( | | ▽ u | | 2 2 ) Δ u + ∫ 0 t g ( t − s ) Δ u ( ⋅ , s ) d s + α u t = 0 , i n Q ^ , where Q ^ is a noncylindrical domain of R n + 1 ( n ≥ 1) with the lateral boundary ∑ ^ and α is a positive constant.

Approaching a vertex in a shrinking domain under a nonlinear flow

We consider here the homogeneous Dirichlet problem for the equation \( u_t = u\Delta u - \gamma |\nabla u|^2 \quad \mathrm{with} \quad \gamma \in R, u \geq 0 \) , in a noncylindrical domain in space-time given by \( |x| \leq R(t) = (T - t)^p, \quad \mathrm{with} \quad p > 0 \). By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x=0, t=T, in the three different cases p>1/2, p=1/2(vertex regular), p<1/2 (vertex irregular).

The heat equation and reflected Brownian motion in time-dependent domains

The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as noncylindrical domains,'' and its connections with partial differential equations. Construction is given for RBM in C3-smooth time-dependent domains in the n-dimensional Euclidean space \Rn. We present various sample path properties of the process, two-sided estimates for its transition density function, and a probabilistic representation of solutions to some partial differential equations. Furthermore, the one-dimensional case is thoroughly studied, with the assumptions on the smoothness of the boundary drastically relaxed.

Compactness of Solutions for Scalar Viscous Conservation Laws in Noncylindrical Domains

We utilize the kinetic formulation approach to study the compactness property for the family {u}>0, solutions of the initial-boundary value problem for the scalar viscous conservation law u t + divxf(u) = xu in a noncylindrical domain. Considering f in C3 and satisfying the non-degeneracy condition, we prove that u is compact in L1loc.

The heat equation and reflected Brownian motion in time-dependent domains.: II. Singularities of solutions

We study the space-time Brownian motion and the heat equation in non-cylindrical domains. The paper is mostly devoted to singularities of the heat equation near rough points of the boundary. Two types of singularities are identified—heat atoms and heat singularities. A number of explicit geometric conditions are given for the existence of singularities. Other properties of the heat equation solutions are analyzed as well.

Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains

We study the initial-boundary-value problems for multidimensional scalar conservation laws in noncylindrical domains with Lipschitz boundary. We show the existence-uniqueness of this problem for initial-boundary data in L ∞ and the flux-function in the class C 1. In fact, first considering smooth boundary, we obtain the L 1-contraction property, discuss the existence problem and prove it by the Young measures theory. In the end we show how to pass the existence-uniqueness results on to some domains with Lipschitz boundary.

Asymptotic behaviour for wave equations with memory in a noncylindrical domains

In this paper we prove the exponential decay astime goes to infinity of regular solutions of the problem for thewave equations with memory and weak damping$u_{t t}-\Delta u+\int^t_0g(t-s)\Delta u(s)ds + \alpha u_{t}=0$ in $\hat Q$where $\hat Q$ is a non cylindrical domains of $\mathbb R^{n+1}$$(n\ge1)$ with the lateral boundary $\hat{\sum}$ and $\alpha$ is apositive constant.

Absolute continuity of parabolic measure and area integral estimates in non-cylindrical domains

We prove the mutual absolute continuity of the parabolic measure associated to divergence form parabolic operators and the surface measure of non-cylindrical domains. This extends the result of [22] on absolute continuity of the caloric measure, with the novelty that, unlike their proof, we do not deal with parabolic layer potentials, but instead, we use the techniques of [20], namely some estimates between area integral and non-tangential maximal function of solutions of the given parabolic equation. This new approach allows us to obtain such mutual absolute continuity for a class of parabolic operators with singular drift terms, as in [18].

-solubility of the Dirichlet problem for the heat equation in non-cylindrical domains

The Dirichlet problem for the heat equation is considered in bounded and unbounded domains of paraboloid type with isolated characteristic points at the boundary. Necessary and sufficient conditions in terms of the weight ensuring the unique solubility of this problem in weighted Sobolev -spaces are found. In particular, a criterion for the solubility of the problem in the classical Sobolev space is established in the case when the domain is a ball.

Dirichlet problems for a nonstationary linearized system of Navier–Stokes equations in non-cylindrical domains

Dirichlet problems for a non-stationary linearized system of Navier-Stokes equations in non-cylindrical domains.

Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains

Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains

Nonlinear wave equation with a nonlinear boundary damping in a noncylindrical domain

Nonlinear wave equation with a nonlinear boundary damping in a noncylindrical domain

The Second Boundary-Value Problem for Pseudoparabolic Equations in Noncylindrical Domains

This paper is devoted to the study of the solvability of the second mixed problem in a noncylindrical domain for the nonstationary equation $${\text{div}}(k(x){\text{ grad }}u_t ) - c(x)u_t - b(x)u(x,t) = f(x,t)$$ called the pseudoparabolic equation. We prove existence and uniqueness theorems for the solution in the case of contracting (as time t increases) domains.

The Non-Homogeneous Wave Equation in Non-Cylindrical Domains

We prove the existence and uniqueness of a global regular solution to the mixed problem for the nonhomogeneous wave equation in a noncylindrical domain using a hiperbolic nonsingular smooth transformation of the original problem to a nonlinear hyperbolic problem in a cylinder.

Time-dependent parabolic problems on non-cylindrical domains with inhomogeneous boundary conditions

We study the relationship between the Dirichlet problem and the Cauchy problem with inhomogeneous boundary conditions for local operators. Our results are applied to non-autonomous parabolic problems on non-cylindrical domains.

A Variational Approach to Evolution Problems with Variable Domains

We solve a parabolic second order partial differential equation in a non-cylindrical domain, under weak assumptions on time-regularity of the boundary; these assumptions are not covered by other methods. We prove an existence result for the solution and an analysis of its regularity is carried over.

Square functions of Calderón type and applications

We establish L^2 and L^p bounds for a class of square functions which arises in the study of singular integrals and boundary value problems in nonsmooth domains. As an application we present a simplified treatment of a class of parabolic smoothing operators which includes the caloric single layer potential on the boundary of certain minimally smooth noncylindrical domains.

Variational Equations of Schroedinger-Type in Non-cylindrical Domains

Cauchy-Dirichlet problems are studied for linear Schroedinger-type partial differential equations in non–cylindrical domains by assuming a monotonicity condition on their sections with respect to time. Existence and uniqueness results are obtained as a consequence of new general results that we prove here for a class of abstract Schroedinger-type differential equations in Hilbert spaces.

Three-dimensional numerical simulation of Marangoni instabilities in non-cylindrical liquid bridges in microgravity

Instability of Marangoni convection in non-cylindrical (convex or concave) liquid bridges of low Prandtl number fluids is investigated by direct three-dimensional and time-dependent simulation of the problem. Body-fitted curvilinear coordinates are adopted; the non-cylindrical original physical domain in the ( r, z, φ) space is transformed into a cylindrical computational domain in a ( ξ, η, φ) space. The geometry of the domain is transformed using a coordinate transformation method by surface fitting technique. The field equations are numerically solved explicitly in time and with a finite difference technique in a staggered grid. The numerical results are analyzed and interpreted in the general context of the bifurcation's theory. The computations show that for semiconductor melts the first bifurcation is characterized by the loss of spatial symmetry rather than by the onset of oscillatory flow and that it is hydrodynamic in nature. The flow field azimuthal organization related to the critical wave number, depends on the geometrical aspect ratio A= L/ D of the liquid bridge and on the shape factor S (convex S>1, concave S<1) of the free surface. The critical azimuthal wave number increases when the geometrical aspect ratio of the bridge is decreased and, for a fixed aspect ratio, can be shifted to higher values by increasing the volume (convex bridges) or to lower values by decreasing the volume (concave bridges). This behavior is explained on the basis of the relation between the typology of the azimuthal disturbances and the structure of the fluid-dynamic field. A generalized law is found to correlate the critical azimuthal wave number of the instability to the geometrical aspect ratio and to the shape factor. A second oscillatory (Hopf) bifurcation occurs when further increasing the Marangoni number. Experimental results available in literature on this second bifurcation are considered for comparison. The experimental and numerical results show a good agreement.