60 research outputs found
High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws
The article first studies the propagation of well prepared high frequency
waves with small amplitude \eps near constant solutions for entropy solutions
of multidimensional nonlinear scalar conservation laws. Second, such
oscillating solutions are used to highlight a conjecture of Lions, Perthame,
Tadmor, (1994), about the maximal regularizing effect for nonlinear
conservation laws. For this purpose, a new definition of nonlinear flux is
stated and compared to classical definitions. Then it is proved that the
smoothness expected in Sobolev spaces cannot be exceeded.Comment: 28 p
Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave
The interactions between linear elastic waves and a nonlinear crack with
finite compressibility are studied in the one-dimensional context. Numerical
studies on a hyperbolic model of contact with sinusoidal forcing have shown
that the mean values of the scattered elastic displacements are discontinuous
across the crack. The mean dilatation of the crack also increases with the
amplitude of the forcing levels. The aim of the present theoretical study is to
analyse these nonlinear processes under a larger range of nonlinear jump
conditions. For this purpose, the problem is reduced to a nonlinear
differential equation. The dependence of the periodic solution on the forcing
amplitude is quantified under sinusoidal forcing conditions. Bounds for the
mean, maximum and minimum values of the solution are presented. Lastly,
periodic forcing with a null mean value is addressed. In that case, a result
about the mean dilatation of the crack is obtained.Comment: submitted to the SIAM J. App. Mat
Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws
Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect for
entropy solutions of nonlinear scalar conservations laws . In this short paper
we will restrict our attention to the simpler one-dimensional case. First,
supercritical geometric optics lead to sequences of solutions
uniformly bounded in the Sobolev space conjectured. Second we give continuous
solutions which belong exactly to the suitable Sobolev space. In order to do so
we give two new definitions of nonlinear flux and we introduce fractional
spaces
Fractional BV spaces and first applications to scalar conservation laws
The aim of this paper is to obtain new fine properties of entropy solutions
of nonlinear scalar conservation laws. For this purpose, we study some
"fractional spaces" denoted , for , introduced by Love
and Young in 1937. The spaces are very closed to the critical
Sobolev space . We investigate these spaces in relation with
one-dimensional scalar conservation laws. spaces allow to work with less
regular functions than BV functions and appear to be more natural in this
context. We obtain a stability result for entropy solutions with initial
data. Furthermore, for the first time we get the maximal smoothing
effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear
degenerate convex fluxes
A two-scale convergence result for a nonlinear scalar conservation law in one space variable
International audienceA general result of two-scale convergence is proven in BV, then it is used for geometric optics expansion near a constant state with two scale for entropy solutions
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