ترغب بنشر مسار تعليمي؟ اضغط هنا

An ill-posedness result for the Boussinesq equation

280   0   0.0 ( 0 )
 نشر من قبل Dan Geba
 تاريخ النشر 2012
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

The aim of this article is to prove new ill-posedness results concerning the nonlinear good Boussinesq equation, for both the periodic and non-periodic initial value problems. Specifically, we prove that the associated flow map is not continuous in Sobolev spaces $H^s$, for all $s<-1/2$.



قيم البحث

اقرأ أيضاً

403 - Luc Molinet 2019
We prove the discontinuity for the weak $ L^2(T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(T) $ as soon as $ s<0 $ and thus completes exactly the well-posedness result obtained by the author.
In this article we present ill-posedness results for generalized Boussinesq equations, which incorporate also the ones obtained by the authors for the classical good Boussinesq equation (arXiv:1202.6671). More precisely, we show that the associated f low map is not smooth for a range of Sobolev indices, thus providing a threshold for the regularity needed to perform a Picard iteration for these problems.
For the Novikov equation, on both the line and the circle, we construct a 2-peakon solution with an asymmetric antipeakon-peakon initial profile whose $H^s$-norm for $s<3/2$ is arbitrarily small. Immediately after the initial time, both the antipeako n and peakon move in the positive direction, and a collision occurs in arbitrarily small time. Moreover, at the collision time the $H^s$-norm of the solution becomes arbitrarily large when $5/4<s<3/2$, thus resulting in norm inflation and ill-posedness. However, when $s<5/4$, the solution at the collision time coincides with a second solitary antipeakon solution. This scenario thus results in nonuniqueness and ill-posedness. Finally, when $s=5/4$ ill-posedness follows either from a failure of convergence or a failure of uniqueness. Considering that the Novikov equation is well-posed for $s>3/2$, these results put together establish $3/2$ as the critical index of well-posedness for this equation. The case $s=3/2$ remains an open question.
We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by the J-self-adjoint linear operator L depending on a small parameter. The problem has been originated from the lubrication approximation of a visc ous fluid film on the inner surface of the rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on the numerical evidence, that the set of eigenvectors of the operator $L$ does not form a Riesz basis in $L^2 (-pi,pi)$. Our method can be applied to a wide range of the evolutional problems given by $PT-$symmetric operators.
135 - Tristan Robert 2018
We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا