Multi-soliton solutions for the supercritical gKdV equations

For the L^2 subcritical and critical (gKdV) equations, Martel proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N solitons asymptotically as time goes to infinity. More recently, for the L^2 supercritical case, Cote, Martel and Merle proved the existence of at least one multi-soliton. In the present paper, as suggested by a previous work concerning the one soliton case, we first construct an N-parameter family of multi-solitons for the supercritical (gKdV) equation, for N arbitrarily given solitons, and then prove that any multi-soliton belongs to this family. In other words, we obtain a complete classification of multi-solitons for (gKdV).Comment: 32 pages, submitted, v2: hyperref links adde

Construction of multi-bubble solutions for the critical gKdV equation

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and $\epsilon_1,\ldots,\epsilon_K\in\{\pm1\}$, there exists an $H^1$ solution $u$ of the equation such that u(t) - \sum_{k=1}^K \frac {\epsilon_k}{\lambda_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{\lambda_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, with $\lambda_k(t)\sim \ell_k t$ and $x_k(t)\sim -\ell_k^{-2}t^{-1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Rapha\"el. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.Comment: 70 page

Contemporary challenges to iodine status and nutrition: the role of foods, dietary recommendations, fortification and supplementation

Iodine deficiency (ID) in women of childbearing age remains a global public health concern, mainly through its impact on fetal and infant neurodevelopment. While iodine status is improving globally, ID is still prevalent in pregnancy, when requirements increase. More than 120 countries have implemented salt iodisation and food fortification, strategies that have been partially successful. Supplementation during pregnancy is recommended in some countries and supported by the WHO when mandatory salt iodisation is not present. The UK is listed as one of the ten countries with the lowest iodine status globally, with approximately 60 % of pregnant women not meeting the WHO recommended intake. Without mandatory iodine fortification or recommendation for supplementation in pregnancy, the UK population depends on dietary sources of iodine. Both women and healthcare professionals have low knowledge and awareness of iodine, its sources or its role for health. Dairy and seafood products are the richest sources of iodine and their consumption is essential to support adequate iodine status. Increasing iodine through the diet might be possible if iodine-rich foods get repositioned in the diet, as they now contribute towards only about 13 % of the average energy intake of adult women. This review examines the use of iodine-rich foods in parallel with other public health strategies, to increase iodine intake and highlights the rare opportunity in the UK for randomised trials, due to the lack of mandatory fortification programmes

Iodine and pregnancy – a qualitative study focusing on dietary guidance and information

Iodine is essential for thyroid hormones synthesis and normal neurodevelopment; however, ~60% of pregnant women do not meet the WHO (World Health Organization) recommended intake. Using a qualitative design, we explored the perceptions, awareness, and experiences of pregnancy nutrition, focusing on iodine. Women in the perinatal period (n = 48) were interviewed and filled in a food frequency questionnaire for iodine. Almost all participants achieved the recommended 150 μg/day intake for non-pregnant adults (99%), but only 81% met the increased demands of pregnancy (250 μg/day). Most were unaware of the importance, sources of iodine, and recommendations for iodine intake. Attitudes toward dairy products consumption were positive (e.g., helps with heartburn; easy to increase). Increased fish consumption was considered less achievable, with barriers around taste, smell, heartburn, and morning sickness. Community midwives were the main recognised provider of dietary advice. The dietary advice received focused most often on multivitamin supplements rather than food sources. Analysis highlighted a clear theme of commitment to change behaviour, motivated by pregnancy, with a desired focus on user-friendly documentation and continued involvement of the health services. The study highlights the importance of redirecting advice on dietary requirements in pregnancy and offers practical suggestions from women in the perinatal period as the main stakeholder group

La fiscalité carbone, facteur de compétitivité. Réduire les charges sociales et énergétiques

National audienceJeudi 22 août, le gouvernement s'est déclaré en faveur d'une " contribution climat-énergie", et déjà les attaques fusent : une nouvelle taxe affaiblirait les entreprises et les consommateurs, alors que les priorités sont la lutte contre le chômage et contre les déficits publics et sociaux, bien plus urgentes que l'évolution du climat. Or, une telle taxe peut précisément répondre à ces priorités. Pour s'en convaincre, il ne faut pas perdre de vue deux forces majeures qui contraignent - et contraindront longtemps - le développement de l'économie française