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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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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 ...
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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 ...
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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 ...
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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+\...
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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?
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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 ...
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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|^...
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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 ...
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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)...
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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 ...
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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 ...
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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 ...
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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 \...