# Questions tagged [uniqueness-theorems]

The uniqueness-theorems tag has no usage guidance.

16
questions

**2**

votes

**2**answers

106 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 ...

**2**

votes

**1**answer

98 views

### 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 ...

**4**

votes

**3**answers

447 views

### Uniqueness of solution to heat equation when initial condition is a generalized function

Let $u(t,x)$ be a solution to the heat equation $$\partial_t = \partial_{xx} \quad (t,x) \in [0,T) \times [-1,1]$$ subject to the initial/boundary conditions
$$u(0,x) = f(x), \quad x \in [-1,1], \\
u(...

**0**

votes

**2**answers

198 views

### Some doubts on proof of pathwise uniqueness of a stochastic differential equation

I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...

**1**

vote

**0**answers

45 views

### Reference request: existence/uniqueness of solutions to convection diffusion equations

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{...

**-1**

votes

**3**answers

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 ...

**2**

votes

**1**answer

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 $f \equiv 0$. ...

**1**

vote

**1**answer

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)\...

**2**

votes

**1**answer

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 ...

**4**

votes

**0**answers

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 ...

**3**

votes

**1**answer

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 \...

**1**

vote

**1**answer

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 ...

**12**

votes

**2**answers

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 ...

**0**

votes

**2**answers

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
& \...

**11**

votes

**1**answer

540 views

### 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 ...

**4**

votes

**2**answers

599 views

### If a PDE has a unique classical solution, must it have a unique viscosity solution?

If a PDE has a unique classical solution, must it have a unique viscosity solution?
The particular problem I am interested in is parabolic, but I would be interested in the general case.
A short ...