Questions tagged [initial-value-problems]

Question 1

I have been taught that for 3D Navier-Stokes (in the precise Clay Math formulation), "bounded energy" ($\int_{\mathbb{R}^n}\lvert u(x,t)\rvert dx<C$) implies existence of $C^\infty$ ...

Question 2

I am somehow confused by "no blowups" condition in Clay Math formulation of Navier-Stokes. Isn't it easy to prove that for a smooth solution the energy can only decrease and therefore it ...

Question 3

Why does the Clay Math problem about Navier-Stokes specifically ask to prove no energy "blowups"? I thought energy inequality has been already proven for $C^\infty$ solutions (and the Clay ...

Question 4

In the book of Roubíček (Nonlinear Partial Differential Equations with Applications) we have Theorem 1.45 basically stating the existence theorem by Carathéodory but we have the linear growth ...

Question 5

I am reading Ordinary Differential Equations and Dynamical Systems by Sideris. The following theorem on the existence of solutions on maximal intervals is stated in the book: Theorem $\mathbf{3.4}:$ ...

Question 6

On page 48 of [Zill2022], there is an initial-value problem (IVP): $$ \frac{dy}{dx}=xy^{1/2}, y(0)=0. \tag{1} $$ I try to solve this IVP by myself. My solution is as follows: Solution: First, it is ...

Question 7

I am working on the following exercise about existence and uniqueness of solutions to initial value problems. Consider the Cauchy problem $$ y'' + |t|^{\alpha}(y')^{2} + \varphi(t)\,|y|^{\beta-1} = f(...

Question 8

I'm trying to solve the following PDE using the Fourier series method: \begin{align} &\partial_t u(t, x) - t\partial^2 u(t, x) = 0 && x \in [0, \pi], \; t\in\mathbb{R}^+ \\ &u(t, 0) = ...

Question 9

Let us consider the following initial value problem $$ \begin{cases} v'(t) = p - f(v)\\ v(0) = 0\\ \end{cases} $$ where $p$ is a positive constant and $f(x)$ is a non negative, unbounded $C^\infty$ ...

Question 10

The given ODE is: $$y'=t^2+e^{-y^2}, y(0)=0.$$ We have to obtain the maximum interval of existence. Now proceeding generally, I got $ M=a^2+1 $ and $ h= \min \left\{ a, \frac{b}{a^2+1} \right\} $. I ...

Question 11

How to show the existence of a solution for a first-order IVP with the slope at the initial value being infinity using the existence theorems. For Eg: $y'(x) = \frac{1}{2\sqrt{x}}, ~y(0)=0,~y'(0)=\...

Question 12

I am looking for an IVP of an ODE without a solution. The examples I found are mostly badly stated, e.g. the initial value is not in the domain or the ODE is not continuous, or they are implicit, e.g. ...

Question 13

Fix constants $C_{1}\neq0,\ C_{2}$, $C_{3}$, and consider the system \begin{align} &(n-1)f'''\Lambda=(n-1)f'f''-(f')^3\tag{\ref{sit_f_1'}}\\ &((n-2)f'f'''-(r_{1}-1)(f'')^2+C_{1}^{2}\...

Question 14

Put 4 equal-mass bodies (planets) on a circle at N S E W, and start them orbiting around their center of mass in a Newtonian gravity field. (There is no central sun, only the mutual 2-body fields.) ...

Question 15

I'm trying to solve the following first-order ordinary differential equation with an initial condition: \begin{cases} y\,dx - \left(4\sqrt{xy} - x\right)\,dy = 0 \\ y(2) = 8 \end{cases} I've tried ...

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Questions tagged [initial-value-problems]

A too easy proof regarding Navier-Stokes [closed]

What is the physical sense of "no blowups" in Navier-Stokes? [closed]

Clarify Navier-Stokes Clay Math problem [closed]

Carathéodory existence theorem and linear growth condition

Solution of ODE exits and reenters compact set finitely many times

How to solve $\frac{dy}{dx}=xy^{1/2}, y(0)=0$?

Existence and uniqueness of solutions for a nonlinear Cauchy problem with $|t|^\alpha$ and $|y|^{\beta-1}$ terms

Problem on choosing the sign of the solution of a PDE using Fourier series

Proving that the solution to an initial value problem is positive for $t > 0$.

Maximum interval of existence and range of solution of the ODE $y'=t^2+e^{-y^2}, y(0)=0.$ [closed]

Proving existence of a solution to an IVP

non-solvable IVP

Proving there exists no solution to a constrained dynamical system

How stable are orbits for the uniform 4-body problem on a circle?

How to solve the first-order ODE with initial condition?

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