# Several Basic Inequalities for PDEs

In mathematics, especially in analysis and PDEs, we evaluate or compare some mathematical quantities in order to reach a desired conclusion. Of course, equality is the best one, however, in many situations we cannot reach our goal only with equalities. … Continue reading

# [M1 Seminar] Week 13 : Elementary Inequalities on the Klainerman Vector Field

We construct the Klainerman vector field with some operators which were defined in the last seminar. And we prove some basic inequalities of these family of operators which play an important role for our final goal. M1_Semi_Week13

# [M1 Seminar] Week 11,12 : Introduction to the Klainerman Vector Field

We introduce the Klainerman vector field and prove some commutator relations which will play an important role in the proof of the global existence theorem for the nonlinear wave equations. M1_Semi_Week11-12

# A Simple Proof of Gronwall’s Inequality

[latexpage] This post introduces a simple proof of Gronwall’s inequality (Thomas Hakon Gronwall, 1877~1932) of the integral form. Let $\alpha, \beta, u \in C^0([a,b])$ with $a<b$. Assume that $\beta \geqslant 0$ on $[a,b]$. Suppose that  u(t) \leqslant \alpha … Continue reading