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Media type:
E-Article
Title:
Oscillatory Solutions to Transport Equations
Contributor:
Crippa, Gianluca;
De Lellis, Camillo
imprint:
Department of Mathematics of Indiana University, 2006
Published in:Indiana University Mathematics Journal
Language:
English
ISSN:
1943-5258;
0022-2518
Origination:
Footnote:
Description:
<p>Let n ≥ 3. We show that there is no topological vector space $\mathrm{X}\subset {\mathrm{L}}^{\mathrm{\infty }}\cap {\mathrm{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1}(\mathrm{\mathbb{R}}\times {\mathrm{\mathbb{R}}}^{\mathrm{n}})$ that embeds compactly in ${\mathrm{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1}$, contains BVloc ∩ L∞, and enjoys the following closure property: If f ∈ Xn(ℝ × ℝn) has bounded divergence and u0 ∈ X(ℝn), then there exists u ∈ X(ℝ × ℝn) which solves $\{\begin{array}{c}{\partial }_{\mathrm{t}}\mathrm{u}+\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{u}\mathrm{f}\right)=0\\ \mathrm{u}(0,\cdot )={\mathrm{u}}_{0}\end{array}$ in the sense of distributions. X(ℝn) is defined as the set of functions u0 ∈ L∞ (ℝn) such that ũ(t,x) := u0(x) belongs to X(ℝ × ℝn). Our proof relies on an example of N. Depauw showing an ill–posed transport equation whose vector field is "almost BV".</p>