Questions tagged [numerical-analysis-of-pde]
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107 questions
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Effective action of unbounded operators on subspaces outside their domain of definition
Consider a densely defined, self-adjoint operator
$$
H: \mathcal{D} \rightarrow \mathscr{H}.
$$
Assume for simplicity that $H$ is nonnegative.
We want to effectively restrict this operator $H$ to a ...
- 113
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0
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58
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Method of characteristics, the characteristic lines follow gradient, is this significant?
The PDE I am working on comes from geology, which I do not have much background on.
Said equation aims to describe describes the erosion by describing it as an advection phenomena: the advection ...
- 11
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0
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93
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Similarity of non-standard matrices
I am researching numerical methods for PDEs. I particular, I am looking at methods for the linear hyperbolic PDE
$$
u_t+au_x=0.
$$
This is a common approach, because successful methods for this model ...
- 111
2
votes
1
answer
62
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Lumped mass matrices and bubble functions for tetrahedral elements
For whatever reason, I stubbornly decided to use tetrahedral elements and find myself needing to use P3 elements with bubble functions ("P3b3d" in FREEFEM-style denomination).
The 2d case is ...
- 981
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45
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Adding a data-dependent term to the porous medium equation while retaining an explicit solution
I am working with the porous medium equation, which I am treating it as a type of Fokker-Planck equation given by:
$
\frac{\partial u}{\partial t} = \Delta(u^m), \quad m > 1
$
For this equation, ...
- 11
4
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198
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Pricing zero coupon bonds through PDE
I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book.
The idea is to model the market price of risk as a ...
- 51
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48
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difference between analytical and numerical solutions for a PDE problem. math problem [duplicate]
first of all I'd like to point out that this is my first time using Stack Overflow so don't hesitate to tell me if something isn't displaying properly or isn't clear.
I'm trying to solve an advection ...
- 9
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21
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Coonvergence rate for the clamped plate problem when approximating with polygonal domains
I'm trying to understand Ridgeway Scott's "A survey of displacement methods for the plate bending problem" [1]. In chapter four he is talking about polygonal approximation and states that ...
- 1
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30
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Peak search for NLS equation solutions
By solving the initial value problem of the NLS equation, I can get a matrix of numerical solutions. I hope to perform a peak search on the matrix of this numerical solution, that is, there are ...
- 33
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99
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Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
- 293
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49
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Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)
What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
- 163
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0
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37
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Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)
Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...
- 111
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0
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36
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Convergence of numerical scheme for HJB equation
Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is:
Consistent
Stable
Monotony
...
- 393
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87
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How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
- 1,759
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134
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Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
- 305
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Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?
Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
- 167
3
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1
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245
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How to generate a random function with conditions?
The background is as follows:
I consider the following differential equation
$$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$
where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...
- 33
1
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0
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292
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Solutions of a Gauss–Codazzi-like system of nonlinear PDEs
Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
- 134
3
votes
2
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428
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Inconsistency in determinability of the solution of a linear first order PDE
Consider the following differential equation:
$$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$
with $u(x,0)=f(x)$. The solution of \eqref{1}, ...
- 350
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2
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928
views
What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
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Studying the evolution of laplacian in NS equation
The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
- 305
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1
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212
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Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?
The split-step method is a numerical method that can be used to solve a nonlinear PDE (https://en.wikipedia.org/wiki/Split-step_method). Even Wikipedia does not refer to the original authors (F.D. ...
- 101
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0
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51
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Error estimates for inhomogeneous semidiscrete PDE
I have the following semidiscrete problem on a meshed domain $U_h$. Let
$V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and
$V_{h\partial}$ be ...
- 235
3
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0
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74
views
Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\partial_t u -a \nabla u - ru (1-u) = 0$$
at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
- 81
1
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1
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106
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How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]
Crossposted at SciComp SE
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find ...
- 31
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0
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30
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Free programs suggestions to simulate parabolic EDPs
I'm interested in learning how to computationally simulate the behavior of parabolic partial differential equations, but I don't know where to start, what are the best free programs to use and where ...
- 677
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395
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Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]
Here it is an Advection-Diffusion equation in 2D:
$$
\frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
2
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0
answers
141
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Approximating solutions to Monge-Ampere from optimal transport plans
I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
- 956
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66
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Computing and isotopy of curves in $\mathbb{R}^3$
Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length.
The (stationary) string can be modelled by a unit speed curve:
$$[0,1] \...
- 171
1
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1
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244
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PDE involving curl
Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE
$$\dfrac{\partial}{\partial t}\...
- 305
4
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2
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566
views
How to compute $\sin(\frac{d}{dx})f(x)$?
Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
- 350
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0
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88
views
Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications
What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...
- 131
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0
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76
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Highy non-linear PDE involving directional derivative
Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed
\begin{equation}\label{ConvoDef}
\left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
- 305
2
votes
2
answers
328
views
$H^s$ norm of non-integer power of functions
Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.
My ...
- 105
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0
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75
views
Flux that can be represented by low and high resolution schemes
In the wiki page of Flux limiter, it writes:
If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
- 111
2
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0
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106
views
Derivation of the Cahn-Hilliard PDE from the point of view of finite difference methods
Consider the Cahn-Hilliard equation
$$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$
defined on your favorite domain. I'm looking for a literature reference that formally ...
- 53
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1
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114
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FEM based solution to parabolic problem
Consider the problem
$$
\begin{cases}
u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega
\end{cases}
$$
...
- 235
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0
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Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition
$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$
\begin{cases}
\partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
- 121
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61
views
How I can distibute values over the computational cells?
I am an engineering student and I try to solve the fluid equations over a given set of computational cells. I have a mathematical question about a field I am currently studying, precisely the ...
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0
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49
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P1-finite element as convolution of P0-finite element
For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...
- 3,425
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How does a computer program recognize shocks given data of a solution to a conservation law?
Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
- 1,755
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164
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Semilinear PDE - BSDE presentation via Feynman Kac formula
For a semilinear PDE, we usually have this FBSDE representation:
$\mathcal{X}_t=\mathcal{X}_0+\int^t_0 \mu (s,\mathcal{X}_s)\, ds\, +\int_0^t \sigma (s,\mathcal{X}_s)dW_s,\quad 0\leq t\leq T, \\
Y_t = ...
- 43
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1
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93
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Typo in error a-priori estimate in a discontinuous Galerkin paper?
I'm looking at this famous paper which is available in the link below:
Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
- 111
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0
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61
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Tuning parameters of PDEs given a set of data
I am interested in doing statistical inference in the context of PDEs. Loosely speaking, the kind of problem I have in mine is the following.
Problem setting
Let $(t_i, x_i, y_i) \in \mathbb{R} \...
- 823
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0
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75
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Dense matrix vs sparse matrix, when they have same number of nonzero elements
I came across a new way in the literature to solve PDE problems numerically, which is called 'Patch Reconstruction'. One example paper is: Li, R., Sun, Z., Yang, F., & Yang, Z. (2019). A finite ...
- 21
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2
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364
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Questions for the non-linear PDE $2u_t=\log(-u_{xx})$
Consider the PDE as follows :
$$2u_t=\log(-u_{xx}), \quad \forall (t,x)\in [0,1)\times (-1,1)$$
with the terminal and boundary conditions
$$u(1,x)=0,\quad \forall -1<x<1 \quad\quad \mbox{and} \...
- 1,334
3
votes
0
answers
89
views
What is the purpose of converting a level-set function into a signed distance function?
In the paper Electrical impedance tomography using level set representation and total variational regularization, the authors tried to implement an iterative algorithm to find the interface of two ...
- 161
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0
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121
views
Smoothly connecting PDEs with finite differences
A PDE with non-smooth inhomogeneity
Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$.
I'm numerically solving the inhomogeneous PDE
\begin{...
- 255
6
votes
1
answer
762
views
What is the big-O complexity of solving the sparse Laplace equation in the plane?
In MATLAB, you can get a 2d Laplacian via A = delsq(numgrid('S',N)); yielding a matrix $A$ that is $n \times n$ with $n = O(N^2)$, for a square domain discretized ...
- 981
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88
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Shape derivative at manifold $M$ in direction $v$ is equal to the shape derivative at $\partial M$ in drection $\langle v,n\rangle n$
Let $\tau>0$ and $d\in\mathbb N$.
Definiton 1$\:\:\:$If $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)\in C^0([0,\tau],\mathbb R^d)$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...
- 167