Category : Mathematics – In Handai

[Differential Equations II] Final Report

[Differential Equations II] Final Report

In this report, we study the global existence of solutions to nonlinear dispersive wave equations begin{eqnarray*}partial_t^2 u + frac{1}{rho^2} |partial_x|^{2rho} u &[...]
[M1 Seminar II] Week 8 : Lorentz Invariance of the Wave Equations

[M1 Seminar II] Week 8 : Lorentz Invariance of the Wave Equations

Now we will introduce some lemmas for the proof of the global existence theorem for the nonlinear wave equations with quadratic nonlinearities. The exsistence theorem whi[...]
[M1 Seminar II] Week 7 : A Global Existence Theorem for Nonlinear Wave Equations

[M1 Seminar II] Week 7 : A Global Existence Theorem for Nonlinear Wave Equations

We consider the following initial value problem [ y_{tt} - Delta y = f(Dy,nabla Dy);;;;text{with};;;; y(t=0) = y_0,; y_t (t=0) = y_1 tag{P} ] with $f in C^infty ( Bbb R[...]
[M1 Seminar II] Week 6 : Weighted a priori Estimates for Small Data

[M1 Seminar II] Week 6 : Weighted a priori Estimates for Small Data

For the proof of the global existence theorem of nonlinear wave equations, this a priori estimates is essential as well as the high energy estimates. Also the high energy[...]
[M1 Seminar II] Week 5 : High Energy Estimates

[M1 Seminar II] Week 5 : High Energy Estimates

We prove the high energy estimates for the nonlinear wave equation on the nonlinearity $alpha =1$. In fact, this problem is a special case of the quasi-linear symmetric h[...]
[M1 Seminar II] Week 4 : Regularities of the Solution and and Improved Existence Theorem

[M1 Seminar II] Week 4 : Regularities of the Solution and and Improved Existence Theorem

We proved the right-continuity of the solution at the initial time in $W^{s,2}$ which implies the continuity of the whole interval. And by the PDE and Sobolev embedding t[...]
[M1 Seminar II] Week 3 : Regularities of the Solution

[M1 Seminar II] Week 3 : Regularities of the Solution

This week we will prove that the solution of the system $u$ is in $C^0([0,T], W^{s,2}) cap C^1([0,T], W^{s-1,2})$ so that $u in C_b^1 ([0,T] times Bbb R^n)$. For this pur[...]
[M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems (2)

[M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems (2)

This week, we will prove the local existence of quasi-linear symmetric hyperbolic systems by using $u^{k}$ which is iteratively defined by the solution of the linear symm[...]
[M1 Seminar II] Week 1 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems

[M1 Seminar II] Week 1 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems

This week, we prove the uniqueness and local existence for quasi-linear symmetric hyperbolic systems using general energy method. M1_Semi2_Week1  [...]
[M1 Seminar] Week 13 : Elementary Inequalities on the Klainerman Vector Field

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

[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 nonlin[...]
[M1 Seminar] Week 10 : Gagliardo-Nirenberg Inequality

[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  [...]
[M1 Seminar] Week 9 : Some Elementary Inequalities

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

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

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

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

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

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

[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[...]