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Regularity and geometric properties of solutions of the Einstein-Vacuum equations

Sergiu Klainerman, Igor Rodnianski (2002)

Journées équations aux dérivées partielles

We review recent results concerning the study of rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. We develop new analytic methods based on Strichartz type inequalities which results in a gain of half a derivative relative to the classical result. Our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates. The latter allows us to take full advantage of the specific structure of...

Regularity and optimal control of quasicoupled and coupled heating processes

Jiří Jarušek (1996)

Applications of Mathematics

Sufficient conditions for the stresses in the threedimensional linearized coupled thermoelastic system including viscoelasticity to be continuous and bounded are derived and optimization of heating processes described by quasicoupled or partially linearized coupled thermoelastic systems with constraints on stresses is treated. Due to the consideration of heating regimes being “as nonregular as possible” and because of the well-known lack of results concerning the classical regularity of solutions...

Regularity and uniqueness for the stationary large eddy simulation model

Agnieszka Świerczewska (2006)

Applications of Mathematics

In the note we are concerned with higher regularity and uniqueness of solutions to the stationary problem arising from the large eddy simulation of turbulent flows. The system of equations contains a nonlocal nonlinear term, which prevents straightforward application of a difference quotients method. The existence of weak solutions was shown in A. Świerczewska: Large eddy simulation. Existence of stationary solutions to the dynamical model, ZAMM, Z. Angew. Math. Mech. 85 (2005), 593–604 and P....

Regularity and uniqueness in quasilinear parabolic systems

Pavel Krejčí, Lucia Panizzi (2011)

Applications of Mathematics

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.

Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity

Andrea C. G. Mennucci (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We formulate an Hamilton-Jacobi partial differential equation H ( x , D u ( x ) ) = 0 on a n dimensional manifold M , with assumptions of convexity of H ( x , · ) and regularity of H (locally in a neighborhood of { H = 0 } in T * M ); we define the “min solution” u , a generalized solution; to this end, we view T * M as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about u ; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number...

Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity

Andrea C.G. Mennucci (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where...

Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity

Alexis Vasseur (2009)

Applications of Mathematics

In this short note we give a link between the regularity of the solution u to the 3D Navier-Stokes equation and the behavior of the direction of the velocity u / | u | . It is shown that the control of Div ( u / | u | ) in a suitable L t p ( L x q ) norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very...

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