Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
3
votes
0answers
38 views
Ratio of solutions to two heat equations
Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
1
vote
0answers
31 views
Weak parabolic maximum principle on Riemannian manifolds
I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken, specifically, I'm reading the ...
0
votes
0answers
22 views
Fundamental solution of parabolic PDE with variable coefficients
Let us consider the parabolic operator
$$
\mathcal{L} = \partial_t - \nabla_x \cdot(a(x)\nabla_x)
$$
over a bounded domain $\Omega\subset\mathbb{R}^d$. The coefficient matrix $a(x)$ is elliptic and ...
1
vote
4answers
166 views
PDE with Laplacian and squared of the gradient
Let $u$ be a real function in $\mathbb{R}^2$. Does anybody know that the following PDE
$$\Delta u+|\nabla u|^2=0$$
has any non-constant general solution or not? It would be appreciated if any one ...
0
votes
0answers
21 views
Approximation in parabolic Sobolev spaces
I am given the following function: fix any $p$ and $\beta \in (0,1)$
$$
u \in L^{p-\beta}(0,T;W^{1,p-\beta}(\Omega)) \quad \text{and} \quad \frac{du}{dt} \in L^{\frac{p-\beta}{p-1}}(0,T; W^{-1,\frac{...
3
votes
0answers
47 views
How to solve this linear Cauchy Problem
within my thesis, I am struggeling with the following PDE:
$u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$
$u(T,x,y)=1,$
where $a,b,c,d,e,f$ are polynomials and the ...
1
vote
1answer
68 views
Steklov averages and negative parabolic sobolev spaces
Suppose one is given a function
$$
w \in L^p(0,T;W^{1,p}(\Omega)) \qquad \text{and} \qquad \frac{dw}{dt} \in L^{p'}(0,T; W^{-1,p'}(\Omega))
$$
I am interested if the following holds:
Denote the ...
2
votes
0answers
74 views
For what functions does Nash inequality becomes equality?
For what functions does Nash inequality becomes an equality? Also any comment on the regularity of these functions (weak solutions to equality)?
Also same question about Poincare inequality.
3
votes
3answers
207 views
Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$
Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$
with initial data $u(0,\cdot) = u_0$.
Fix $\alpha >0$. How can I estimate (without using explicitly ...
2
votes
0answers
152 views
A question about whether an operator can be lipschitz or not
Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$.
Now define the operator $ \mathcal{A} : C^{\sigma, \sigma/2}(X) \to C^{\sigma, \...
3
votes
0answers
82 views
Isometries along the normalized Ricci flow
As we know the Ricci flow preserves isometries of the initial manifold along the flow. But I want to know does the normalized Ricci flow preserves isometries of the initial manifold along the flow as ...
1
vote
1answer
166 views
Classification of a system of two second order PDEs with two dependent and two independent variables
If we have a second order quasilinear PDE of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
0
votes
1answer
43 views
Boundary controllability of the heat equation and observation
I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...
6
votes
0answers
142 views
Curvature decay of Ricci expanders
Let $M$ be a gradient Ricci expander with nonnegative curvature operator. Assume $\Sigma$ is its space of directions at infinity (so $M$ looks like a cone over $\Sigma$).
What is the curvature ...
0
votes
1answer
44 views
Classical solution of one dimensional Parabolic equation and a priori estimates
I am researching a system of pdes and it leads me to study the classical solution of one dimensional linear parabolic equation: $u_t+Lu=f,\, t\in[0,T],x\in \Omega=[0,1]$, where $L$ is non-divergence ...
1
vote
0answers
74 views
Heat equation, free boundary and dynamic programming
I have a dynamic programming problem with an underlying diffusion $$ d X_t = \mu \, dt + d b_t$$
where $b_t$ is a standard brownian motion.
The HJB equation for the value function $v(x,t)$ I get is ...
1
vote
1answer
62 views
Reference request: Existence/regularity for viscous Hamilton-Jacobi equations
A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as:
\begin{equation*}
u - \epsilon \Delta u + H(Du) = f(x)
\end{equation*}
or
\begin{equation*}
u_{t} -...
2
votes
0answers
73 views
Observability inequality for the heat equation
I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:
\begin{array}{c}
\varphi _{t}+\...
2
votes
0answers
34 views
Existence (linear) parabolic second order system in non-divergence form
I am looking for a hint to a reference: I am dealing with a system of parabolic equations of the form
$$
\partial_t u_i=\sum_{j=1}^m a_{ij}(x,t)\Delta u_j,\quad i=1,\ldots,m
$$
set on a open and ...
1
vote
0answers
46 views
Reference request for Vanishing viscosity method paper
I am trying to find the paper/book where they study problem (1),(3) given bellow using Vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.
A little bit of ...
1
vote
0answers
69 views
Backward heat equation [closed]
I have seen many countre examples concerning the instability and the ill-posedness of the backward heat equation, but all these examples are done in the $||.||_{\infty}$. My questions are:
1) Is the ...
0
votes
0answers
45 views
Vanishing viscosity limits in PDEs and random functions
The vanishing viscosity method consists in viewing problem
$$(A) \hspace{1cm}
u_t+g(u)_x = 0\\[2ex]
$$
as the limit of the problem
$$(B) \hspace{1cm}
u_t+g(u)_x+\nu \varDelta u = 0\\[2ex]
$$...
3
votes
1answer
377 views
$L^p$-norm under the heat flow
Let $(M, g)$ be a compact Riemannian manifold.
Assume that $u_0$ is a positive smooth function on $M$ and let $u_t = e^{t \Delta} u_0$ be the solution to the heat equation on $(M, g)$ with initial ...
5
votes
2answers
273 views
Functional decaying under the heat flow (?)
Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$.
For any positive function $v$, I set
$$
J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g.
$$
...
2
votes
1answer
100 views
Rate convergence of the heat equation as diffusion tends to zero
Is there a good reference for the following problem? Consider any smooth bounded domain $\Omega$ and solve the heat equation
\begin{align}
\partial_t u^\kappa &= \kappa \Delta u^\kappa,\\
u^\...
2
votes
2answers
282 views
Reference for De Giorgi-Nash-Moser theory
I am interested in Holder regularity for equations of the form
$$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash ...
1
vote
1answer
91 views
schauder regularity heat equation
Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...
0
votes
0answers
69 views
Hölder continuity of weak subsolutions to parabolic equations
It is well known that a weak solution to linear parabolic equation
$u_t - div(A(x)\nabla u) = 0$ with homogeneous Neumann boundary condition, is Holder continuous with $A(x)$ belongs only to $L^\infty(...
1
vote
1answer
72 views
Vorticity equation for generalized Naiver Stokes equations
In $3D$ or $2D$, I can get the vorticity equation for the incompressible NSE; however, what's the vorticity equation for the generalized NSE? Does the fractional laplacian commute with the curl?
0
votes
0answers
58 views
Estimate of $\|\nabla u\|_{L^{\infty}(\mathbb R^2)}$ for incompressible Navier-Stokes equations
Consider the incompressible Navier-Stokes equations on $\mathbb R^2:~u_t - \Delta u + u\cdot\nabla u + \nabla p=f$. I want to estimate $\|\nabla u\|_{L^{\infty}(\mathbb R^2)}$. Is there a way to ...
1
vote
0answers
52 views
Parabolic (heat) PDE Green's function spatial asymptote at infinity
Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation, with the diffusion coefficient dependent on the spacial variables. Does its ...
1
vote
0answers
135 views
Critical spaces and energy estimate in NS equation [closed]
There is a ‘rescaling transformation’ that is particularly significant for
the Navier–Stokes equations when they are posed on the whole space, but is
also important in the local regularity theory.
...
2
votes
0answers
82 views
Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space
It is known that for $\alpha\in(0,1)$ and $p>1$,
the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by
$$
W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
1
vote
0answers
49 views
Inhomogeneous heat kernel estimates
I am looking for existence results on inhomogeneous linear heat equations. Concretely, I have the equation
$$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x), ~~~~~u(0, x) = u_0(x)$$
where $\...
1
vote
1answer
109 views
Expected properties for a PDE whose solution is supposed to be something that doesn't exist
My understanding of Lecture #33, 34: The Characteristic Function for a Diffusion:
As an alternative to directly computing the characteristic function of a random variable $X_t$ in a stochastic ...
1
vote
0answers
75 views
Reference request: $|\partial_t u - \Delta u|\in L^p(\Omega^T)$ implies Holder estimate
I am currently reading a paper regarding harmonic flow between Riemannian manifolds. Let $\Omega$ be a Riemannian domain of dimension $2$, $\Omega^T=\Omega\times[0,T]$. The equation is
$$
\partial_t ...
4
votes
0answers
80 views
Linear PDE with non constant coefficients and properties of Green's Function
Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...
3
votes
1answer
137 views
Parabolic Regularity with Neumann B.C
Consider the parabolic problem in the cylinder of base $B$, the unit ball,
$$
\partial_t u -\text{div}\left( A(x) D u +F(t,x)\right)=0 \text{ in } (0,T)\times B,
$$
with $(ADu +F)\cdot \nu=0$ on $(0,T)...
0
votes
0answers
35 views
Looser condition for regularity for Neumann problems
If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then
$$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$
Let's say we are studying rotationally symmetric solutions to ...
6
votes
1answer
181 views
Injectivity of a Fredholm operator
While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.
Question : Suppose $K$ is ...
1
vote
0answers
86 views
Looking for access to McKean's original paper?
I'm looking for the PDF version/scan of Henry P McKean Jr.'s paper on propagation of chaos. The reference is as follows -
Propagation of chaos for a class of non-linear parabolic equations., In ...
3
votes
0answers
61 views
Reference request : maximal regularity for the heat equation on the torus
Fix $d\geq 1$ and $1<p,q<+\infty$. I am searching for a reference concerning the following result.
There exists a constant $C=C(d,p,q)$ such that, for any
$u\in\mathscr{C}^\infty(\mathbb{R}...
0
votes
0answers
67 views
Movement of a random walk in the limit (a particle in diffusion)
I asked this question in Math Exchange and obtained no answer.
Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
1
vote
1answer
131 views
Motivation behind the parabolic metric
I've been reading some papers about parabolic evolution problems between manifolds. We want to study the behaviour of maps from the domain $(0,T)\times \Omega$, where $\Omega\subset \Bbb R^m$, to some ...
0
votes
1answer
122 views
Green's functions/fundamental solution to a non-constant coefficients pde
We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
1
vote
0answers
63 views
Differentiable dependence on the data for parabolic equations
Let $g_{\lambda}$ be a one parameter family of Riemannian metrics, which are complete and with bounded curvature, on the unit disk, depending smoothly on the parameter $\lambda$. Let $\Delta_{\lambda}$...
3
votes
0answers
76 views
One-dimensional harmonic map flow with low regularity
My question is the following:
What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in ...
1
vote
1answer
91 views
Generator of spacetime Markov Process is Parabolic?
Suppose one considers some non-autonomous SDE thereby the Markov transition function is not homogeneous. In order to "recover" some homogeneity, one can consider the "spacetime" or "lifted" ...
1
vote
0answers
119 views
Classical solutions for parabolic PDE's
I am studying Ricci flow theory and the Ricci flow is not a parabolic equation. But there is some variant of it, called DeTurck-Ricci equation, that happens to be a parabolic PDE. So to argue ...
0
votes
0answers
35 views
Regular Approximation for Degenerate Parabolic PDE
Let $T>0$ and $\Omega \subset \mathbb{R}^N$ be a bounded domain with smooth boundary. I'm considering a problem of porous medium equation
$$
\partial_t u -d \Delta u^m = F(u) \ in \ (0,T) \times \...
