Rough Initial Data Research Articles

Analysis of fully discrete approximations to parabolic problems with rough or distribution-valued initial data

We consider fully discrete approximations to a parabolic initial-boundary value problem with rough or distribution-valued initial data in two space dimensions. For discretization in time and space, we apply single step methods and the standard Galerkin method with piecewise linear test functions, respectively. For spatial discretization of the initial condition, we are however forced to use more involved constructions. Our main result is stability and error estimates of the discrete solutions.

RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION OF LIQUID CRYSTALS

We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.

Finite Element Approximation with Quadrature to a Time Dependent Parabolic Integro-Differential Equation with Nonsmooth Initial Data

In this paper we analyze the effect of numerical quadrature in the finite element analysis for a time dependent parabolicin tegro-differential equation with nonsmooth initial data. Both semi-discrete and fully discrete schemes are discussed and optimal order error estimates are derived in L ∞ (L 2 ) and L ∞ (H 1 ) norms using energy method when the initial function is only in H 1 0 . Further, quasi-optimal maximum norm estimate is shown to hold for rough initial data.

The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order

In this paper we study some dissipative versions of the Korteweg de Vries equation with rough initial data. By working in Bourgain's type spaces we prove local and global well posedness results in Sobolev spaces of negative order. In particular, we prove the global well posedness of the Korteweg de Vries Burgers equation u t + u x x x - u x x + uu x = 0 in H s (R) for s > - 34/ - ½4.

The optimal control of nonlinear diffusion equations with rough initial data

We consider in this paper a problem consisting of the optimal boundary control of a nonlinear diffusion equation with a “rough” initial condition, that is, in which the initial condition is a (unit) mass at an (interior) point of its domain. Since the state space may need to be unbounded, it is necessary to study this problems by means of the techniques of nonstandard analysis. Firstly, a problem is transformed into a semi-infinite linear programming problem by embedding the spaces of admissible trajectory-control pairs into spaces of measures. Then this is mapped into an appropriate nonstandard structure, where a near-minimizer is found for the nonstandard optimization; this entity is mapped back as a minimizer for the original problem. An appendix is including introducing the basic concepts of nonstandard analysis.

On the Strong Solvability of the Navier—Stokes Equations

In this paper we study the strong solvability of the Navier—Stokes equations for rough initial data. We prove that there exists essentially only one maximal strong solution and that various concepts of generalized solutions coincide. We also apply our results to Leray—Hopf weak solutions to get improvements over some known uniqueness and smoothness theorems. We deal with rather general domains including, in particular, those having compact boundaries.

Maximum Norm Analysis of Completely Discrete Finite Element Methods for Parabolic Problems

We consider full discretizations of linear parabolic initial value problems in any space dimension. Finite elements are used for the discretization in space. The resulting semidiscrete problem is integrated in time by means of a strongly $A(\theta )$-acceptable rational method. Stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data.

Parabolic Equations for Curves on Surfaces: Part II. Intersections, Blow-up and Generalized Solutions

We describe a theory for parabolic equations for immersed curves on surfaces, which generalizes the curve shortening or flow-by-mean-curvature problem, as well as several models in the theory of phase transitions in two dimensions. A class of equations is described for which the initial value problem is well-posed for rough initial data, for which one can give a description of the way a smooth solution becomes singular, and for which one can define generalized solutions, i.e., solutions which are smooth, except at a discrete set of times. The methods which are used in this paper are more geometrical than those of Part I. By comparing arbitrary solutions with certain special solutions, and by considering the way they intersect, we derive estimates for the curvature and the tangent, which allow one to study the initial value problem, and the way solutions become singular.

On the Backward Euler Method for Parabolic Equations with Rough Initial Data

The backward Euler method is applied for the discretization in time of a general homogeneous parabolic equation in weak form. A short proof is given that, with k the time step, the norm of the error at time t is bounded by $Ckt^{ - 1} $ times the norm of the initial data. The result permits application to equations which are already discretized in space.

Geometric flows with rough initial data

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in $L^\infty$ and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball.

Blow up and regularity for fractal Burgers equation

The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian α < 1/2, and global existence as well as analyticity of solution for α ≥ 1/2. We also prove the existence of solutions with very rough initial data u0 ∈ Lp, 1 < p < ∞. Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.

Full discretization of the porous medium/fast diffusion equation based on its very weak formulation

SKA ‡ Abstract. The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H 1. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown even for rough initial data and right-hand sides. The theoretical results are illustrated for the piecewise constant finite element approximation of the porous medium equation with the�-distribution as initial value. As a byproduct, L p -stability of theH 1 -orthogonal projection onto the space of piecewise constant functions is shown. ut − �(juj p−2 u) = f; p >1; u= u(x;t); u(�;0) = u0; (x;t) 2 � (0;T); whereR d is a nice bounded domain, −� is the Laplace operator acting on the spatial variables, u0 are given initial data, f is a given right-hand side and we assume some boundary condition on the boundary of . We immediately see that for p = 2 this is the classical heat equation. For p > 2, it is referred to as the porous medium equation. In this case u is the density of the gas at a given point and time. For 1 2, in the one dimensional case, the solution is given by (5.2). We will later use this in numerical experiments. The solution is also known in higher dimensions, see again Vazquez (22, Chapter 4). Throughout the article, we focus on the following generalization of the above