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Questions tagged [partial-differential-equations]

Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
Elias Costa's user avatar
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94 votes
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3k views

Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
hase_olaf's user avatar
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28 votes
0 answers
493 views

Wave equation: predicting geometric dispersion with group theory

Context The wave equation $$ \partial_{tt}\psi=v^2\nabla^2 \psi $$ describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is ...
Sal's user avatar
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25 votes
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2k views

The heat kernel as a distance metric on manifolds

I recently came across Varadhan's formula (see e.g. [1], [2], [3], [4], [5]): $$ {d_{\text{g}}(x,y)^2}{} = -\lim_{t \rightarrow 0} 4 t \log K_t(x,y) $$ where $d_\text{g}$ is the geodesic distance ...
user3658307's user avatar
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24 votes
0 answers
646 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
user avatar
23 votes
0 answers
464 views

Level sets of solution to Nonlinear PDE

I work on Stochastic Control theory and BSDE's for my research. In my research, I characterized the set I am interested in as the level set of a function which is a viscosity solution to nonlinear PDE ...
chandu1729's user avatar
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17 votes
1 answer
436 views

Proving that $\int u^2 dx < Ce^{-at}$ where $u$ solves a linear pde

Let the unit sphere in $\mathbb{R^n}$ be $B_1(0)$ and let $u$ be the smooth solution of $$ \begin{cases} u_{tt} + a^2(x) u_t - \Delta u = 0 & B_1(0) \times (0,\infty)\\ u(x,t) = 0 & \partial ...
Merkh's user avatar
  • 3,640
17 votes
0 answers
298 views

Is $\frac{1}{H(x) \pm i0}$ a distribution if $|\nabla H| \neq 0$ for $H(x)=0$?

I know that $\frac{1}{x \pm i0}$ is a tempered distribution in $\mathcal{S}'(\mathbb{R})$, see e.g. the Sokhotski–Plemelj theorem. In some lecture notes online I found the following statement (without ...
user avatar
16 votes
0 answers
280 views

Regularity of parabolic PDEs for large $\lambda$

Let $\Omega$ be a sufficiently smooth domain, $T>0$, and $L$ be the following elliptic operator of the divergence form: $$Lu(t,x)=a^{ij}(t,x)u_{ij}(t,x),$$ such that $a^{ij}\in C([0,T]\times\bar{\...
John's user avatar
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16 votes
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The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
jef808's user avatar
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15 votes
0 answers
533 views

Solving PDE on manifold via Hodge theory

Let $(M, g)$ be a Riemannian manifold, where $M$ is compact without boundary. The Hodge decomposition tells us that $$\Omega^k = \ker (\Delta) + \text{Im} \ d + \text{Im}\ d^* . $$ Note that we can ...
user avatar
14 votes
0 answers
566 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
booksee's user avatar
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13 votes
1 answer
365 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
Rajesh D's user avatar
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13 votes
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How weird can the boundary be so that the fundamental theorems of vector calculus hold?

Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
BigbearZzz's user avatar
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13 votes
0 answers
1k views

Schilling's proof of the Feynman-Kac Formula for Brownian motion

This is part of a proof to the Feynman-Kac formula from Schilling's Brownian motion. I need some help understanding the proof to this theorem. Theorem (Kac 1949). Let $(B_t)_{t\ge 0}$ be a $d$-...
nomadicmathematician's user avatar
13 votes
0 answers
1k views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\,\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)\,x\right]\!\psi(x,t) $$ where $...
xslittlegrass's user avatar
13 votes
1 answer
350 views

What does "smooth solution" of Ricci flow mean?

A way to formalize smoothness of a flow is to think of the spacetime, see e.g. here. Let's say we are flowing a compact manifold $M$. Which of the following is true? $g_{ij}$ is a $C^1$ in time and $...
Igor Belegradek's user avatar
13 votes
1 answer
522 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
Montaigne's user avatar
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12 votes
0 answers
413 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^...
Svetoslav's user avatar
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12 votes
0 answers
247 views

Question about boundedness of a sequence in $ W^{3,q} $ for any $ 1\leq q < \frac{N}{N-1} $

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can Calderón-...
Fin8ish's user avatar
  • 3,361
11 votes
0 answers
279 views

What is the interpretation/intuition of $e^{itA}$ for a self-adjoint unbounded operator?

Let $A : i \frac{d}{dt} : D(A) \to H^1([0,1])$ with domain $D(A)=H^1_*([0,1])=\{u \in H^1([0,1]): u(0)=u(1)\} \subseteq H^1([0,1])$. Then I know that $A$ is self-adjoint. Using the spectral theorem, ...
Suspicious Fred's user avatar
11 votes
0 answers
165 views

Noether's theorem in the critical heat equation

I am watching a serie of lectures on "Blow up solution for the energy critical heat equation" from Monica Musso on YouTube and at some point she states a result I do not understand. Let met ...
Falcon's user avatar
  • 4,044
11 votes
0 answers
167 views

An inequality about quasi-linear function

Let $\gamma$ be a positive, nondecreasing, continuous, function defined on $[0,\infty]$. Suppose that $\gamma(x+y)\le C(\gamma(x)+\gamma(y))$. In addition, suppose $$ \int_{2}^{\infty}\frac{dr}{\gamma(...
Mr.xue's user avatar
  • 681
11 votes
0 answers
428 views

Representation theoretic explanation of Huygens' Principle

Huygens' principle -- Wave equations in $\mathbb{R}^n$ seem similar for different $n$, but behave quite differently depending on the parity of $n$: waves in odd dimensional spaces never look back, ...
Student's user avatar
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11 votes
0 answers
784 views

Numerically solving a non-linear PDE by an ODE on the Fourier coefficients

I need to solve numerically a PDE of the form $$ u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x) $$ with initial condition $u(x,0)=u_0(x)$. I can assume that both $u(\cdot,t)$ and $a(\cdot)$ are ...
epsilone's user avatar
  • 441
11 votes
0 answers
305 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
yess's user avatar
  • 1,002
11 votes
0 answers
256 views

What classical conditions give unique Laplace equation solutions on a half-plane?

Solutions of the Laplace equation on the upper half plane of $\mathbb{R}^{2}$ are not unique: $$ \frac{\partial^{2}}{\partial x^{2}}u+\frac{\partial^{2}}{\partial y^{2}}u = 0,\;\;\;...
Disintegrating By Parts's user avatar
11 votes
2 answers
2k views

I don't understand the 'idea' behind the method of characteristics

Below is an image of my lecture notes explaining the idea behind the method of characteristics for quasilinear first order PDEs. However I don't understand how the curve $C_s$ is defined and how it ...
user53076's user avatar
  • 1,614
10 votes
0 answers
407 views

The Heat Equation in Brezis' book

I am reading the heat equation in Functional Analysis, Sobolev Spaces, and Partial Differential Equations by Haim Brezis, and having some concerns about the proof, whose screenshot is as attached ...
Justin Lien's user avatar
10 votes
0 answers
320 views

Convergence of numerical methods for Viscous Burgers' Equation

For linear problems we can deduce convergence of numerical schemes by checking consistency and stability because of the Lax Equivalence Theorem. For conservation laws, we know that conservative, ...
428's user avatar
  • 565
10 votes
0 answers
517 views

Heat equation proving smoothness

I have a question regarding a PDE course: Let $T$ be the strongly continuous semigroup which belongs to the heat equation, thus with generator $A$ is the Laplacian. Suppose we have $g \in C^{\infty}...
nippon's user avatar
  • 435
10 votes
2 answers
422 views

Uniqueness of solutions to $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$

The problem I am working on is to show that there is a unique compactly supported solution to the PDE $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$, $(x, t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0)= \...
user187437's user avatar
10 votes
0 answers
969 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
Cristopher Moore's user avatar
10 votes
0 answers
272 views

When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar, cylindrical or spherical coordinates ...
nabla's user avatar
  • 1,279
9 votes
0 answers
713 views

Forward vs backward formulation in Feynman-Kac

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a nice filtered probability space with an $m$-dimensional standard Brownian motion $W$. Fix a time horizon $T>0$. Let $\mu \colon [0,T] \times \mathbb{R}^d \...
Florian R's user avatar
  • 1,267
9 votes
0 answers
504 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 ...
user avatar
9 votes
0 answers
165 views

Elliptic regularity on the Hypercube

Assume $$ Lu=f\quad \text{in } [0,1]^d\\ u=0 \quad\text{ on } \partial[0,1]^d $$ for some strongly-elliptic operator $L$, and $f\in H^k$$, k\geq -1$. Do we have $u\in H^{k+2}$? I can only find the ...
Bananach's user avatar
  • 7,984
9 votes
1 answer
299 views

Density given by variable-coefficient PDE

I am looking for a time-dependent probability density $f(x,y,t)$ solving the equation $$-\frac{\partial f}{\partial t} = \alpha\cdot \big(y - F(x)\big)\frac{\partial f}{\partial x}+\beta\cdot \big(G(y)...
day1pnl's user avatar
  • 121
9 votes
0 answers
165 views

Solving a 2nd-order elliptic PDE with non-constant coefficients

I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$: $$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$ I notice that the l.h.s. is the ...
Theon Alexander's user avatar
9 votes
0 answers
623 views

Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. However, I learn also from the conversations in the same post that the ...
Ali Taghavi's user avatar
9 votes
0 answers
221 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
Nikolaj-K's user avatar
  • 12.3k
9 votes
0 answers
277 views

$W^{1,p} $ and $W^{2,p}$ Estimates.

In the beginning of section 4 in here the author says that one can easily adapt the methods in the preceding section to obtain $W^{1,p}$ estimate. I'm trying to do this. I think the following: the ...
user29999's user avatar
  • 5,271
8 votes
0 answers
108 views

Gronwall lemma with highly oscillatory kernel

As a toy model for a larger problem, I want to show that if $A,B\geq 0$ and $u(t)$ satisfies $$|u(t)|\leq A+\left|\int_0^t B\cos(s^2)u(s)ds\right|$$ then $u$ satisfies a bound like $$|u(t)|\leq AC$$ ...
kieransquared's user avatar
8 votes
0 answers
108 views

Smooth tiling of the plane

For my master thesis, I solved a PDE under the assumption of the domain being smooth and small. I wanted to patch these domains and solutions somehow together, hoping that I can get a global result. ...
Eric's user avatar
  • 1,130
8 votes
0 answers
571 views

Langevin equation and convergence to stationary solutions. Free energy. SDE. FPE.

Let $f\geq 0$ be Lipschtiz. The overdamped Langevin equation \begin{equation}\label{eq overdamped Langevin SDE} dX=-\nabla f(X)dt+\sqrt{2} dW_t \end{equation} with Kolmogorov forward equation \...
orange is the new f's user avatar
8 votes
0 answers
156 views

Solutions to Monge–Ampère equation

I am looking for examples of solutions to the following Monge-Ampere equation (also called a determinant Hessian PDE, and note that this is analogous to finding a surface given its Gaussian curvature) ...
Nick P's user avatar
  • 223
8 votes
0 answers
335 views

Validating solution to PDE using integral transforms

I'm trying to obtain the analytical solution of a Fokker-Planck PDE, which the solution is a probability density function, and then use this to find the mean of some quantity in the paper. The paper ...
rami_salazar's user avatar
8 votes
0 answers
317 views

Extension of Burgers' equation

I recently encountered a viscous Burgers' equation type PDE, but with the addition of a derivative-squared nonlinear term (in dimensionless form): $u_t - u_{xx} + uu_x - u_x^2 = 0\,,$ where the ...
JonasB's user avatar
  • 71
8 votes
0 answers
323 views

Gagliardo–Nirenberg–Sobolev inequality for weighted Sobolev space with exponential weights

Consider the weighted $L^p_\omega(\mathbb{R}^d)$ space on $\mathbb{R}^d$ be the set of Lebesgue measurable functions such that $$\|f\|_{L^p_\omega}=\int_{\mathbb{R}^d}|f|^p\omega_\mu(x)\,dx< \...
John's user avatar
  • 13.3k
8 votes
1 answer
461 views

Proving that system is Hamiltonian

I am trying to show that the PDEs governing stratified flow are Hamiltonian. The approach is based on the paper "Nonlinear Stability Analysis of Stratified Fluid Equilibria which can be found here ...
Master's user avatar
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