Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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### 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, ...

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**0**answers

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 ...

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**0**answers

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**

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**4**answers

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

**0**answers

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{...

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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 ...

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vote

**1**answer

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 ...

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**0**answers

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.

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votes

**3**answers

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 ...

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votes

**0**answers

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, \...

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votes

**0**answers

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

**1**answer

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
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...

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votes

**1**answer

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(...

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votes

**0**answers

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 ...

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votes

**1**answer

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**

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**0**answers

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

**1**answer

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

**0**answers

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}+\...

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votes

**0**answers

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

**0**answers

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 ...

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**0**answers

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 ...

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votes

**0**answers

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

**1**answer

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 ...

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votes

**2**answers

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

**1**answer

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^\...

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votes

**2**answers

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

**1**answer

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+\...

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votes

**0**answers

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(...

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vote

**1**answer

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?

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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 ...

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**0**answers

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 ...

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**0**answers

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.
...

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votes

**0**answers

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|^...

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vote

**0**answers

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

**1**answer

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

**0**answers

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 ...

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votes

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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 ...

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**0**answers

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 ...

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votes

**0**answers

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}...

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votes

**0**answers

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 ...

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vote

**1**answer

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 ...

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votes

**1**answer

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 ...

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**0**answers

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 $\Delta_{\lambda}$...

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**0**answers

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

**1**answer

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" ...

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**0**answers

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 ...

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**0**answers

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 \...