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Questions tagged [parabolic-pde]

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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0answers
118 views

Interior gradient estimate of mean curvature equation

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. I'm stuck in the theorem $2.3$. ...
0
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0answers
69 views

Linearization around a traveling wave

In [1], Remark 2.1., the authors say the following: ".. in moving coordinates, the linearization of (37) around a traveling wave profile is given by \begin{equation} v_t = x \partial^2_x v + \frac{2}{...
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2answers
76 views

Reference request for fractional Poincare inequality

Suppose we consider in $\mathbb R^n$, then how to show $\Vert f \Vert_{L^{p}} \leq C\Vert \nabla^{s}f \Vert_{L^{q}}^{\alpha}$, where $s>0$ is noninteger and $\alpha \in (0,1)$?
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0answers
42 views

Weak Maximum Principle For Parabolic Equations With Neumann Boundary Condition

Consider the problem: \begin{equation} Lu\equiv \sum^n_{i, j=1} a_{ij}(x, t)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum^n_{i=1} b_i(x, t)\frac{\partial u}{\partial x_i}+c(x, t)-\frac{\partial u}...
1
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1answer
81 views

Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: \begin{equation} C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T). \end{equation} what condition should be put ...
3
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0answers
81 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, ...
6
votes
1answer
199 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 ...
1
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0answers
33 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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4answers
176 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
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0answers
22 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
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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
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1answer
69 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
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0answers
76 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
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3answers
210 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
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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
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0answers
84 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
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1answer
173 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
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1answer
52 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
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0answers
144 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
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1answer
45 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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0answers
77 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
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1answer
68 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
74 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
37 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
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0answers
62 views

Reference request for a paper with Vanishing viscosity method and smooth approximation of initial data

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
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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
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0answers
59 views

Vanishing viscosity limits in PDEs and random perturbations

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
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1answer
379 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
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2answers
274 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
101 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
315 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
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1answer
95 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
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0answers
70 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
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1answer
73 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
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0answers
61 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
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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
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0answers
85 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
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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
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1answer
112 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
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0answers
77 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
91 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
141 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
39 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
185 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
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0answers
93 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
65 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
68 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
141 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
139 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 ...