22 research outputs found
Traveling wave solutions for the FPU chain: a constructive approach
Traveling waves for the FPU chain are constructed by solving the associated
equation for the spatial profile of the wave. We consider solutions whose
derivatives need not be small, may change sign several times, but decrease
at least exponentially. Our method of proof is computer-assisted. Unlike other
methods, it does not require that the FPU potential has an attractive
(positive) quadratic term. But we currently need to restrict the size of that
term. In particular, our solutions in the attractive case are all supersonic
On the constants in a basic inequality for the Euler and Navier-Stokes equations
We consider the incompressible Euler or Navier-Stokes (NS) equations on a
d-dimensional torus T^d; the quadratic term in these equations arises from the
bilinear map sending two velocity fields v, w : T^d -> R^d into v . D w, and
also involves the Leray projection L onto the space of divergence free vector
fields. We derive upper and lower bounds for the constants in some inequalities
related to the above quadratic term; these bounds hold, in particular, for the
sharp constants K_{n d} = K_n in the basic inequality || L(v . D w)||_n <= K_n
|| v ||_n || w ||_{n+1}, where n in (d/2, + infinity) and v, w are in the
Sobolev spaces H^n, H^{n+1} of zero mean, divergence free vector fields of
orders n and n+1, respectively. As examples, the numerical values of our upper
and lower bounds are reported for d=3 and some values of n. Some practical
motivations are indicated for an accurate analysis of the constants K_n.Comment: LaTeX, 36 pages. The numerical values of the upper bounds K^{+}_{5}
and K^{+}_{10} for d=3 have been corrected. Some references have been
updated. arXiv admin note: text overlap with arXiv:1009.2051 by the same
authors, not concerning the main result
Computer-assisted proofs for radially symmetric solutions of PDEs
We obtain radially symmetric solutions of some nonlinear (geo-
metric) partial differential equations via a rigorous computer-assisted method.
We introduce all main ideas through examples, accessible to non-experts. The
proofs are obtained by solving for the coefficients of the Taylor series of the
solutions in a Banach space of geometrically decaying sequences. The tool that
allows us to advance from numerical simulations to mathematical proofs is the
Banach contraction theorem
Computer-assisted proofs for radially symmetric solutions of PDEs
We obtain radially symmetric solutions of some nonlinear (geo-
metric) partial differential equations via a rigorous computer-assisted method.
We introduce all main ideas through examples, accessible to non-experts. The
proofs are obtained by solving for the coefficients of the Taylor series of the
solutions in a Banach space of geometrically decaying sequences. The tool that
allows us to advance from numerical simulations to mathematical proofs is the
Banach contraction theorem
INTERVAL INCLUSION COMPUTATION FOR THE SOLUTIONS OF THE BURGERS EQUATION
National Natural Science Foundation of China [10571146]In this paper we study the interval computation for the solutions of the Burgers equation. For the initial-boundary value problems of the Burgers equation by using the technique of the Green function, a new kind of interval method is proposed. Both algorithm and computational examples are given. Convergence is proved. From the results we see that this interval method can get a better solution with our corroboration
On the constants in a Kato inequality for the Euler and Navier-Stokes equations
We continue an analysis, started in [10], of some issues related to the
incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus
T^d. More specifically, we consider the quadratic term in these equations; this
arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two
velocity fields. We derive upper and lower bounds for the constants in some
inequalities related to the above bilinear map; these bounds hold, in
particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v
. D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity)
and v, w are in the Sobolev spaces H^n, H^(n+1) of zero mean, divergence free
vector fields of orders n and n+1, respectively. As examples, the numerical
values of our upper and lower bounds are reported for d=3 and some values of n.
When combined with the results of [10] on another inequality, the results of
the present paper can be employed to set up fully quantitative error estimates
for the approximate solutions of the Euler/NS equations, or to derive
quantitative bounds on the time of existence of the exact solutions with
specified initial data; a sketch of this program is given.Comment: LaTeX, 39 pages. arXiv admin note: text overlap with arXiv:1007.4412
by the same authors, not concerning the main result