[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   Related Posts ?[M1 Seminar] Week 13 : Elementary … Continue reading

[M1 Seminar] Week 10 : Gagliardo-Nirenberg Inequality

We show the standard Gagliardo-Nirenbergy inequality just by using some elementary inequalities and induction. M1_Semi_Week10   Related Posts ?[M1 Seminar] Week 9 : Some Elementary InequalitiesSeveral Basic Inequalities for PDEs[M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric … Continue reading

[M1 Seminar] Week 9 : Some Elementary Inequalities

We introduce some elementary inequalities which will be used for proving the Gagliardo-Nirenberg inequality. M1_Semi_Week9   Related Posts ?[M1 Seminar] Week 10 : Gagliardo-Nirenberg Inequality[M1 Seminar] Week 8 : Some Inequalities with Friedrich Mollifiers[M1 Seminar] Week 13 : Elementary Inequalities … Continue reading

[M1 Seminar] Week 8 : Some Inequalities with Friedrich Mollifiers

We introduce some elementary inequalities with Friedrich mollifiers which will be frequently used later. M1_Semi_Week8   Related Posts ?[M1 Seminar] Week 9 : Some Elementary Inequalities[M1 Seminar] Week 13 : Elementary Inequalities on the Klainerman Vector Field[M1 Seminar] Week 10 … Continue reading

[M1 Seminar] Week 6,7 : A Global Existence Theorem to the Linear Symmetric Hyperbolic Systems

This week, we prove the global existence theorem to the linear symmetric hyperbolic system by using the standard energy inequality. M1_Semi_Week6-7   Related Posts ?[M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems (2)[M1 Seminar II] … Continue reading

[M1 Seminar] Week 5 : Energy Estimates for the Linear Symmetric Hyperbolic System

We derive the corresponding estimates for higher derivatives from the basic energy estimates which was given in the last week. We also prove a simple corollary using this result. And we discuss about the finite propagation speed of the hyperbolic … Continue reading

[M1 Seminar] Week 3,4 : Linear Symmetric Hyperbolic System

   This week, we derive basic energy estimates for the linear symmetric hyperbolic system. M1_Semi_Week3-4   Related Posts ?[M1 Seminar] Week 5 : Energy Estimates for the Linear Symmetric Hyperbolic System[M1 Seminar] Week 6,7 : A Global Existence Theorem to the … Continue reading

[M1 Seminar] Week 2 : Decay Estimates for Linear Wave Equations

   This week, we have considered only odd space dimensions for the linear wave equation to prove two estimates in Theorem 2.1. The proof was done for the simple case. For the general cases including the case of even dimensions, we … Continue reading

[M1 Seminar] Week 1 : Introduction to Nonlinear Wave Equations

   For nonlinear wave equations, we cannot guarantee that the solutions exist for some initial value problems. Also the global solutions may exist but also it may have some singularities depending on the initial conditions. Thus we need the global existence … Continue reading

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