# Questions tagged [heat-equation]

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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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### 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 ...
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### The heat kernel in Hermitian bundles over Riemannian manifolds

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....
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### Heat kernel on Riemannian manifold

The idea to construct a heat kernel is first construct a parametric in a small neighbourhood. Then use a bump function to extend it. And do convolution iteratively. (Reference: Laplacian on a ...
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### Quick question on the constants involved in heat kernel upper bounds

Let $M$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the Laplace-Beltrami operator on $M$. It is known that for small $t$, let's say, $0< t < t_0(M, g)$, the heat ...
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### One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
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### Gradient blowup for the 1-dim heat equation near an irregular boundary point

I'm trying to estimate the rate of boundary gradient blow-up for the 1-dim heat equation near an irregular boundary point. Let $b(t) := (1-t)^\alpha$, $\alpha < 1/2$. Let $u(t,x)$ solve the ...
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### Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
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### Heat kernel and convergence

Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...
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### 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 ...
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### 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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### 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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### Heat kernel asymptotic expansion on complete noncompact manifold

I was wondering if there is any reference for the asymptotic expansion of heat kernel on a complete noncompact manifold. That is, \begin{align} H(x,q,t) \sim \frac{e^{-\frac{d^2(q,x)}{4t}}}{(4\pi t)^{...
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 $(0,T)... 0answers 87 views ### Reference request : maximal regularity for the heat equation on the torus Fix$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}...
Is anything known about the Laplacian on squashed spheres $S^{2n-1}_\omega$, where the ambient $C^n$ coordinates satisfy $$1= \sum_{i=1}^n \omega_i |z_i|^2$$ for fixed real numbers $\omega_i$? for ...