# Questions tagged [uniqueness-theorems]

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### 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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I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form $$\frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{... 3answers 106 views ### Are <sum, product, N> triplets unique and hard to solve? [closed] This question comes from some reasoning I made myself about a "joke block chain" where every new block is labeled with a triplet <S, P, N> where where S = sum of the N transactions so ... 1answer 71 views ### Unique continuation of the Hilbert transform Let's consider the usual Hilbert transform H defined as$$Hf = P.V. (\frac{1}{x}*f).$$A well-known unique continuation principle states that if Hf = f =0 on some interval I, then f \equiv 0. ... 1answer 73 views ### Uniqueness of function with range \mathbb{S}^2 under a constraint Assume g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2 are two bijective functions defined on the set A. Now assume a constraint C: \forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\... 1answer 98 views ### Reference request: uniqueness for a certain PDE systems I'm working on a system of the following form:$$(1) \,\,\,\,\,\ \begin{cases} u_{tt} + L_1u + L_2v= f, \\ \nabla u - \nabla v - \nabla v_t=0 \end{cases} $$where u(x,t) and v(x,t) belong to ... 0answers 101 views ### Ricci flow on locally symmetric noncompact manifold As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ... 1answer 148 views ### Uniqueness of minimizers in the Calculus of Variations Let f \colon \mathbb R^2 \to \mathbb R be the function defined by$$ f(x,y):= (x^+)^2 + (y^+)^2 $$where a^+ = \max\{a,0\} for any real number a. Given a Lipschitz regular domain \Omega \... 1answer 111 views ### Lotka Volterra existence of Caratheodory solution I strive to prove that the following system of differential equations:$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$has a unique Caratheodory solution ... 2answers 1k views ### Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid? This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold: https://arxiv.org/abs/1709.10033 What's the current ... 2answers 102 views ### Uniqueness problem for an elliptic system I want to prove the uniqueness of the solution of the following problem:$$\eqalign{ & - d\,\Delta u + u = {u^p} \text{ in } \Omega \cr & u > 0 \text{ in } \Omega \cr & \...
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 ...