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} \...
12 votes
2 answers
613 views

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....
0 votes
1 answer
83 views

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 ...
0 votes
0 answers
52 views

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....
0 votes
1 answer
232 views

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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3 votes
1 answer
228 views

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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7 votes
1 answer
537 views

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, ...
0 votes
0 answers
34 views

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 ...
2 votes
2 answers
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 ...
2 votes
1 answer
121 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
1k 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(...
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0 votes
2 answers
240 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
50 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{...
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-1 votes
3 answers
127 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
89 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 $f \equiv 0$. ...
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1 vote
1 answer
76 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
109 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
110 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
180 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 \...
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1 vote
1 answer
121 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 ...
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13 votes
2 answers
2k 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 ...
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0 votes
2 answers
109 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 & \...
  • 525
14 votes
1 answer
643 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
666 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 ...
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