Local Existence Of Classical Solutions Research Articles

Local existence of classical solutions for the Einstein–Euler system using weighted Sobolev spaces of fractional order

We prove the existence of a class of local in time solutions, including static solutions, of the Einstein–Euler system. This result is the relativistic generalisation of a similar result for the Euler–Poisson system obtained by Gamblin (1993). As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article. To cite this article: U. Brauer, L. Karp, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

On the dual systems to Euler and Navier-Stokes equations in R3

In this paper we consider two model systems of equations obtained by taking the Hodge duals of the Euler and the Navier–Stokes systems, respectively. The ‘vorticity’ formulation of the new system becomes a nonlinear, non-local scalar equation having similarities to the quasi-geostrophic equations. We prove the local existence of classical solutions, and derive the Beale-Kato-Majda type of blow-up criterion for the dual Euler system. Moreover, we actually show that the finite-time blow-up occurs for generic non-trivial initial data for this system. For the dual Navier-Stokes equations we construct an explicitly global-in-time smooth solution for general initial data, using the Hopf-Cole transform. In the appendix we formulate the Euler system and the dual Euler system in terms of the Clifford algebra.

Gradient rearrangement for diffeomorphisms of a compact manifold

Dealing with smooth diffeomorphisms on a compact riemannian manifold, we recast in differential geometric terms the results of Brenier and McCann on optimal mass transportation via gradient rearrangement, which lack of a regularity theory. We proceed to a pde approach of the gradient rearrangement, proving uniqueness and local existence of classical solutions; we reduce global existence to a priori estimates (left open, except near flat metrics). We discuss the link between factorization of diffeomorphisms and the Helmholtz decomposition of vector fields, including a new result on the Moser–Ebin–Marsden factorization. A nonlinear comparison principle of independent interest is established.

On the Cauchy Problem for a Nonlinear Kolmogorov Equation

We consider the Cauchy problem related to the partial differential equation Lu ≡ ∆xu + h(u)∂yu − ∂tu = f (· ,u ), where (x, y, t) ∈ R N × R × )0 ,T (, which arises in mathematical finance and in the theory of diffusion processes. We study the regularity of solutions regarding L as a perturbation of an operator of Kolmogorov type. We prove the existence of local classical solutions and give some sufficient conditions for global existence.

Time Evolution of Nematic Liquid Crystals with Variable Degree of Orientation

We consider an evolution system, describing the time-dependent behavior of nematic liquid crystals with variable degree of orientation within the continuum model of Ericksen. We establish a dissipation relation and prove both the global existence of weak solutions and the local existence of classical solutions. Furthermore, we investigate the stability and long-time behavior of solutions and obtain an exact solution of the corresponding stationary system in a one-dimensional case.

INITIAL-BOUNDARY VALUE PROBLEM AND CAUCHY PROBLEM FOR A QUASILINEAR EVOLUTION EQUATION

In the present paper, the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation utt-Δut=∑i=1n∂∂xiσi(uxi)are studied under the assumptions that do not require the monotonicity of σi(s) (i = 1, …, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.

Local existence of classical solutions to first-order parabolic equations describing free boundaries

Local existence of classical solutions to first-order parabolic equations describing free boundaries

Heat flow with tangent penalisation converging to mean curvature motion

In this paper, we prove that mean curvature motion can be regarded as the singular limit of the following model:where ε > 0 is a small parameter and . This model is related to the Landau–Lifshitz equation of ferromagnetism. Local existence of classical solutions of the Dirichlet problem and global existence of the travelling wave solutions are also obtained.

Global wellposedness for quasi‐linear symmetric hyperbolic systems with regularized coefficients

AbstractFollowing the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi‐linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local‐in‐time convergence to the classical solution of the non‐regularized problem is proved.

The Existence and Uniqueness of Solutions of Equations for Ideal Compressible Polytropic Fluids

The local existence of classical solutions for a characteristic initial boundary value problem for the equations of ideal compressible polytropic fluids is proved. The problem is replaced by a system of well–posed problems and then the method of successive approximations is used.

Local existence of classical solutions to the Landau equations

Abstract In this paper we prove the local existence of classical solutions to the Landau equations. A smallness condition on the data at infinity is required. We use the maximum principle and Bernstein's method to obtain the boundedness of solutions and an approximate scheme is given.

The solvability of the equations of motion of mechanical systems with sliding friction

The problem of the existence of local classical solutions of the equations of motion of holonomic systems with sliding friction is investigated. The conditions for the equations of motion to be solvable for the accelerations enable estimates to be obtained of the coefficients of friction, within the framework of which (from the point of view of the existence of solutions) Coulomb's laws can be used to describe the dynamics of mechanical systems with sliding friction.

Les solutions à support compact de l’équation du mouvement des atmosphères d’étoiles

The initial-boundary value problem for the equation of movement of the atmosphere of a star is discussed with compactly supported initials. To establish the local existence of classical solutions, a suitable symmetrization and the concept of “gentle solutions”, are introduced. It is further shown that the life span is finite for any non-trivial “gentle” solution with support compact.

Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum

Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum