# Questions tagged [uniqueness-theorems]

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### Uniqueness of global solution

I am reading Section 3.3 of this paper, and trying to understand the proof of uniqueness of a global solution to the following equation defined on the Torus $\mathbb{T}^3$ \begin{align*} \mathrm{d} \...
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### Definitions of determinant by unique features

A well-known definition of the determinant is: The determinant is the only function of a vector space of dimension $n$ to its underlying field which is multilinear, alternating and normalized. See e....
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### Fokker-Planck: uniqueness and convergence to stationary distribution

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
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### Implicit function theorem when $dF/dy = 0$ but under monotonicity constraint of the implicit function $y(x)$

I am looking for an extended version of the implicit/inverse function theorem that would show uniqueness of a strictly increasing implicit function, even when the derivative condition is violated (e.g....
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### Proof: If a reproducing kernel exists for a Hilbert space, then it is unique

I really want to prove the statement in the title but I'm struggling with it. Here my current state: Proof via contradiction. Let $\mathcal{H}$ be a RKHS with two reproducing kernels $k$ and $\hat{k}$ ...
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### A simple question on the Navier-Stokes system

The Navier-Stokes system for incompressible fluids in $\mathbb R^3$ reads as \begin{align} &\frac{\partial v}{\partial t}+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0, \quad \text{div} v=...
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### Looking for an electronic copy of Holmgren's old paper

I would like to know if anyone has an electronic copy of the following paper: "Holmgren, E.: Über Systeme von linearen partiellen Differentialgleichungen. Översigt Vetensk. Akad. Handlingar 58, ...
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### Existence and Uniqueness of lifting Hele-Shaw problem

I am researching for the existence and uniqueness of solutions for the equation in figure below enter image description here $$\nabla\cdot u = \frac{\dot b(t)}{b(t)} \text{ in }\Omega(t) \tag{1}$$ The ...
203 views

### Is the converse of Osgood criterion for ODEs also true?

Namely, Assuming that $f$ is a continuous real function and $f(0)=0$ , $f(x)>0$ when $x\neq 0$, Consider the differential equation $x'= f(x)$ with the initial value $x(0)=0$ , is it true that if ...
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### What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?

I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the ...
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### Existence and uniqueness of geodesics in low regularity

Consider a Riemannian manifold $(M,g)$. How much regularity is required of $g$ so that for any $x\in M$ and $v\in T_xM$ with $|v|=1$ there exists a unique geodesic \$\gamma\colon(-\epsilon,\epsilon)\to ...
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