An Elementary Solution of the Navier-Stokes Existence and Smoothness: Elementary Weak Solutions

This is an application of the functional analysis to existence and smoothness of the Navier-Stokes equations using elmentary weak solutions in Sobolev spaces. There are no long or complicated calculations, semi-groups or apriori estimates or boundary conditions on <i>Ω</i>' ⋐ℝ^<em>3</em> are not used at all. We apply the locally solvability of linear partial differentiatial operators with constant coefficients, erase the pressure p, and solve<i>∂u/∂t －△u＝</i> <i>Pf－P(u</i>_<em>n</em><i>・▽)u</i>_<em>n</em>_,<i>∂u/∂t</i> －△<i>u</i> ＝ <i>Pf－P(u・▽)u</i>in <i>X</i>＝∩ _{<em>m ≥5, p=1, 2</em>}_{<em> </em>}<i>W</i>^<em>m,p</em><i>(Ω)</i> , <i>Ω</i> ⋐ℝ×ℝ³.<i>Ω </i>is an open set in ℝ×ℝ³ ＝ℝ⁴, for example, four-dimensional open ball. We handle the variables <i>t, x</i><b> </b>simultaneously. The non-zero examples of elements of <i>X </i>do not depend on choice of <i>c'</i>_{<em>k, m</em>} .For example,|| (<i>x</i>^{<em>i </em>}<i>→ </i>sin <i>x</i>^<em>i</em> )(<i>x</i>^{<em>j </em>}→ cos <i>x</i>^{<em>j </em>})||_<em>X</em>≤_{<em> </em>}<i>C ||</i>sin||_{<em>X </em>}<i>||</i>cos||_<em>X</em> , for <i>i, j </i>＝1, 2, 3,||∂_{<em>j </em>}^{<em>k </em>}exp ||_{<em>X </em>}≤ <i>C</i>^{<em>k </em>}<i>|| </i>exp ||_{<em>X ,</em>}||∂_{<em>j </em>}^{<em>k </em>}sin ||_{<em>X </em>}≤ <i>C</i>^{<em>k </em>}<i>|| </i>sin ||_<em>X</em> , and||∂_{<em>j </em>}<i>u</i>_{<em>n </em>}<i>||</i>_<em>X</em><i> </i>≤ <i>CM </i>even if <i>n</i>≥2, so by mathematical induction,||<i>u</i>_{<em>n+1 </em>}<i>－ u</i>_{<em>n </em>}<i>||</i>_{<em>X </em>}≤_{<em> </em>}<i>L</i>^{<em>n </em>}<i>||u</i>_{<em>n </em>}<i>－ u</i>_{<em>n－1 </em>}<i>||</i>_{<em>X ,</em>}therefore, existence follows similar to the proof of Banach's fixed point theorem. The solutions <i>u,</i><i> </i>p satisfy<i>∂u/∂t －△u</i> ＝ <i>f－▽</i>p<i>－(u・▽)u</i>in the sense that distributions in <i>X</i>_<em>σ</em>_{<em> ,</em>}div<i> </i><i>u </i>＝0in the sense that distribution in <i>X</i>_<em>.</em><i>f ≠ </i>0 ⇒ <i>u </i>≠ 0, <i>f</i>＝0 ⇒ <i>u</i>＝0. There exists the non-zero solution by Fujita, Kato or Shibata even if <i>f </i>＝0, so elementary weak solutions are different from the solutions by Fujita, Kato or Shibata.Our weak solutions do not satisfy the energy inequality on <i>Ω</i>' ⋐<i>ℝ</i>³, but our solutions are smooth, unique, <i>u</i> is defined on ℝ×ℝ³ , and,lim ∂^α<i>u(t,</i> <i>x</i>) ＝0 ( <i>t</i>, |<i>x</i>|→∞)for any multi-index α. The map <i>f</i>→<i>u</i> is continuous. The initial value<i>a</i>∈<i>C</i>^<em>∞</em><i>(Ω')</i>∩<i>L</i>^<em>p</em><i>(Ω'),</i>our solutions<i>u</i>∈<i>C(I;</i> <i>L</i>^<em>p</em><i>(Ω')).</i>If∫₀ ^<em>T</em>(<i>f</i>(<i>s</i>), <i>φ</i>(<i>s</i>))_{<em>L2(Ω')</em>}<i>ds</i> + (<i>a</i>, <i>φ</i>(0))_{<em>L2(Ω')</em>}_{<em> </em>}＝0and<i>u</i>|_∂Ω＝0, then <i>f</i>＝0, <i>a</i>＝0, <i>u</i>＝0 satisfy the energy inequality and <i>u</i> belongs to the Serrin class, so <i>u</i> is the unique Leray-Hopf's weak solution and an elementary weak solution, therefore, if <i>u</i><i> </i>is the Leray-Hopf's weak solution and an elementary weak solution, then <i>u</i>＝0.Japanese edition: https://note.com/reviewer_amzn_m/n/n68e3f2c95d74 . There are many introduction links to mathematical analysis in Japanese on the article.We studied in x.com , mathlog.info , note.com and Tokyo University of Science. Many people helped us to research from 2/5/2022 to 6/29/2025.<i>∂</i>_{<em>j </em>}: <i>S→X </i>is a closed operator, so <i>S </i>is a closed set, therefore the uniqueness follows.If we change the definition of <i>S </i>:<i>S</i> ＝ { <i>u</i>∈ <i>X </i>: ||<i>u</i>||_{<em>X </em>}≤ <i>M , </i>∀<i>k</i> , ||<i>∂</i>_{<em>j </em>}^{<em>k </em>}<i>u ||</i>_{<em>X </em>}≤_{<em> </em>}<i>C</i>^<em>k</em> ||<i>u</i>||_{<em>X </em>}},then the existstence follows exactly.