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Suppose we have a function $u(x,y,z,t)$ satisfies the following weak form: $$ \partial_t u + u\cdot abla u + abla p = 0,\ \ \mathrm{div} u = 0,\ (x,y,z) \in \Omega \times \mathbb{R}^2 \times \mathbb{R}$$ $$ u = 0 \ \ \mathrm{on} \ \partial \Omega \times \mathbb{R}^3 \times \mathbb{R} \\ u(x,y,z,0) = u_0(x,y,z) \ \ \mathrm{in}\ \Omega \times \mathbb{R}^3 \times \mathbb{R}. $$ And, suppose we have the $L^2$ norm $||u_0||_{L^2(\Omega \times \mathbb{R}^3)} = 1$, $$ L^2(\Omega \times \mathbb{R}^3) = \left\{u_0 \in L^2(\Omega \times \mathbb{R}^3)\ :\ ||u_0||_{L^2(\Omega \times \mathbb{R}^3)} = 1\right\}, $$ $$ \langle u_0,u_0 \rangle_{L^2(\Omega \times \mathbb{R}^3)} = ||u_0||_{L^2(\Omega \times \mathbb{R}^3)}. $$ How can I estimate the $L^2$ norm of the solution $u$? I want to apply the so-called Lions' Lemma to $u$. Thanks. A: Consider $f$ defined on the $n+1$ variables $x \in \Omega$, $y \in \mathbb{R}^n$, $z \in \mathbb{R}$, and $t \in \mathbb{ What's New in the? in the years since the agreement, an agreement from which Japan has benefited, did not specify that the "war deterrent" was to be removed once the West punished Japan for its failure to respect the Neutrality Treaty. [3] For convenience, the court will only discuss the West and the United States. For Japan, the situation is identical. [4] The court is not aware of any precedent from any jurisdiction that requires the state's government or its political subdivisions to assume the financial consequences of violations of a law for which the state has waived its immunity. The corollary, of course, is that the State cannot evade liability simply by voluntarily waiving its immunity for a defined set of violations. [5] This holding is not intended to be final on the questions of whether the United States can be sued for monetary relief when it breaches its treaty obligations and whether the United States' waiver of sovereign immunity for violations of the Non-Aggression clause affects the United States' ability to be sued for monetary relief. Image copyright AFP/Getty Images Image caption The sculptures adorn the walls of the capital's Emperor's Park China's Forbidden City, which inspired the Sydney Opera House, is to have a new $165m (£118m) facelift. The three-tiered Ming-dynasty complex in Beijing has been undergoing a renovation for some years now. But the work, officially due to be finished by the end of 2015, has been delayed. Now it is expected to be completed by 2018. China's president Xi Jinping is fond of playing up the country's long history and heritage, and the Forbidden City complex has a special role. There is also a duty, of course, in what it can tell us about China itself, but that will not be the main focus here. We will confine ourselves to the art, both ancient and modern, for which the palace is now most famous. The Forbidden City is a wordplay on "forbidden city" - the name for a place like Beijing's Imperial Palace that has been off limits for citizens for much of its history. Image copyright Getty Images Image caption The name "Forbidden City" derives from the area's former status as the emperor's residence But the palace is more than that - it is also a spiritual place. Its name itself suggests not just the former status of its inhabitants and the privileges and restrictions they lived with. The palace System Requirements For Photoshop 2021 (Version 22.4.2): Windows 7/8.1/10. X64 only. 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