All Questions
Tagged with ap.analysis-of-pdes uniqueness-theorems
15 questions
0
votes
0
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86
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+100
Uniqueness of bubbling points in Struwe's global compactness theorem
I am reading the following paper of Struwe in which he proves the following result:
Proposition 2.1:
Let $n\geq 3$, $\lambda \in \mathbb{R}$ and $\Omega$ be a smoothly bounded domain in $\mathbb{R}^{n}...
- 537
0
votes
0
answers
46
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Uniqueness results for linear first order systems of PDEs
Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$):
$$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
- 1,205
2
votes
0
answers
152
views
Uniqueness of the solution to systems of first-order linear PDEs
Context:
Let $\Omega \subset \mathbb{R}^p$ be an domain.
For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
- 51
5
votes
0
answers
233
views
Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
- 1,075
3
votes
0
answers
102
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Uniqueness continuation property for parabolic equation
Consider the following parabolic equation:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial \rho }{\partial t}-\Div\left( a\left( x\right) \nabla
\rho \right) +p(x)\rho = 0 & \...
1
vote
0
answers
82
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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} \...
- 101
3
votes
1
answer
308
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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=...
- 16.2k
7
votes
1
answer
597
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, ...
- 509
2
votes
1
answer
169
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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 ...
- 313
3
votes
3
answers
2k
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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
$$u(0,x) = f(x), \quad x \in [-1,1], \\
u(...
- 103
1
vote
0
answers
94
views
Reference request: existence/uniqueness of solutions to convection diffusion equations
I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form
$$
\frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{...
- 111
2
votes
1
answer
118
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 ...
- 21
13
votes
2
answers
3k
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 ...
- 233
0
votes
2
answers
119
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
& \...
- 617
4
votes
2
answers
829
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 ...
- 383