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Time Existence (time + existence)

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Existence of a weak solution to the Navier,Stokes equation in a general time-varying domain by the Rothe method

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2009
í Neustupa
Abstract We assume that ,t is a domain in ,3, arbitrarily (but continuously) varying for 0,t,T. We impose no conditions on smoothness or shape of ,t. We prove the global in time existence of a weak solution of the Navier,Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q[0,,T) := {(x,,t);0,t,T, x,,t}. The solution satisfies the energy-type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2004
Karol Mikula
Abstract A new method for solution of the evolution of plane curves satisfying the geometric equation v=,(x,k,,), where v is the normal velocity, k and , are the curvature and tangential angle of a plane curve , , ,2 at the point x,,, is proposed. We derive a governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves and prove local in time existence of a classical solution. These equations include a non-trivial tangential velocity functional governing a uniform redistribution of grid points and thus preventing numerically computed solutions from forming various instabilities. We discretize the governing system of equations in order to find a numerical solution for 2D anisotropic interface motions and image segmentation problems. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Non-linear initial boundary value problems of hyperbolic,parabolic type.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002
A general investigation of admissible couplings between systems of higher order.
This is the second part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so-called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right-hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non-linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Non-linear initial boundary value problems of hyperbolic,parabolic type.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2002
A general investigation of admissible couplings between systems of higher order.
This is the third part of an article that is devoted to the theory of non-linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so-called energy method. In the above sense the regularity assumptions about the coefficients and right-hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Power concavity on nonlinear parabolic flows

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2005
Ki-Ahm Lee
Our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p -Laplace equation, called power concavity, and that the power is determined by the homogeneity of the parabolic operator. In the parabolic p -Laplace equation for the density u, the concavity of u(p,2)/p is considered, which indicates why the log-concavity has been considered in heat flow, p = 2. In addition, the long time existence of the classical solution of the parabolic p -Laplacian equation can be obtained if the initial smooth data has -concavity and a nondegenerate gradient along the initial boundary. © 2004 Wiley Periodicals, Inc. [source]