Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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1answer
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Diffusion equation problem

Heat conduction is described by the diffusion equation $$u_t = k u_{xx},$$ where $u(x,t)$ is the temperature at $x$ at time $t$ and $k$ is some constant. A homogeneous thin pipe occupying the region ...
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1answer
13 views

Differentiation and PDE Theory

I have been given the following two definitions: 1) $D^ku$ is the set of all derivatives of order k of u 2) Let $\Omega$ be a non-empty subset of Euclidean space $\mathbb{R}^N.$ An expression of the ...
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0answers
16 views

Elliptic boundary value problems and elliptic partial differential equations

I am interested in the relation between the definition of an 'elliptic boundary value problem' and an 'elliptic partial differential equation'. From the wiki entries it seems that 'elliptic boundary ...
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0answers
22 views

Weak solution of elliptic equation depends continuously on parameter

Suppose I have a weak formulation of the form: find $u \in H^1_0(\Omega)$ such that $$\int_\Omega b(p)(\nabla u \nabla v + \lambda uv)=0$$ holds for all $v \in H^1_0(\Omega)$ where $b:[a,b] \to ...
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0answers
2 views

what is the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE?

I want to know the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE when right hand side of the elliptic operator is $L^1.$ Also how does right hand side with ...
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0answers
23 views

Essay about PDEs

I've taken an introductory course in PDEs and I have to write an essay of 4-8 pages on a topic in partial differential equations. The topics we touched on are: First order linear partial ...
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0answers
8 views

Solving the PDE's naturally arising from requiring vector fields to commute

If one has two vector fields $X$ and $Y$ in $\mathbb{R}^4$ whose coefficients are indeterminate functions (some coefficients are linearly related to others, however), then the condition that the ...
2
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1answer
593 views

weak solution of biharmonic equation

Consider $U$ a open and bounded subset of $R^n$, with smooth boundary. A weak solution for the problem : $$ \Delta^2 u = f \ \in \Omega \ and \ u=\frac{\partial u}{\partial\nu} = 0 \text{ in } ...
2
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1answer
25 views

Divergence structure equation

Consider Laplace's equation with potential function $c$: $$-\Delta u + cu = 0, \tag{$*$}$$ and the divergence structure equation $$-\operatorname{div}(aDv)=0, \tag{$**$}$$ where the function $a$ is ...
2
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1answer
113 views

Null Lagrangians and “Local Degree”

Let $u: U\subset\mathbb R^n \rightarrow \mathbb R^n$ be a smooth function, $U$ bounded. Let $x_0$ and $r$ be such that $B_r(x_0)$ is disjoint from $\partial U$. Let $\eta$ be a smooth bump function ...
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0answers
11 views

diffusion equation [on hold]

I'm kinda lost with this problem. I don't know how to solve it. If somebody can help me I will be so thankfully. I'm so confuse.If somebody know a reference problem that would help a lot
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1answer
24 views

Two density results, $C_{0}^{\infty}$ is dense in $L^{2}$

I really lack analysis background but I'm in a situation with PDE's (Finite elements specifically) where I'm trying to prove a couple intermediate results. Prove $C_{0}^{\infty}(\Omega)$ dense in ...
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0answers
22 views

heat equation, total heat energy [duplicate]

I'm having a hard time with this problem. I get the situation, but I just don't know how to model it and show part b and part c. I will be so thankfully.
0
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1answer
22 views

Boundary conditions

I am kinda confuse with the second part of my homework. I did the first part (3/a and 4/a) without any problem, but part b for both problems I don't get it at all. I try to plug the boundaries in the ...
2
votes
1answer
38 views

Weyl asymptotic law

In Panoramic view in Riemannian geometry of Berger, I met the following formula $$\sum e^{-\lambda_i t} \sim \frac{\vert \Omega\vert }{2\pi t} -\frac{\vert \partial \Omega\vert}{\sqrt{2\pi t}} + ...
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0answers
45 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
2
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1answer
19 views

Eigenfunctions of the laplacian (1 dimension)

I have the following problem: $\frac{d^2 u}{dx^2}(x)+\lambda u(x)=0, x \in (a,b)$ and $u(a)=u(b)=0$. The general solution (for $\lambda>0$) is $u(x)=c_1\cos(\sqrt\lambda x)+c_2 \sin (\sqrt\lambda ...
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0answers
22 views

PDEs, infinite vector spaces, and functional analysis

Suppose I have a PDE for some quantity $q_{t}$:$= q(\boldsymbol{x},t)$ at some time $t$ on a 2D vector space $\boldsymbol{x} = (x_{1},x_{2})$, that looks, for the sake of the question, something like ...
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1answer
11 views

Question about Convergence Definition for Finite Difference Scheme

I have a question about the Convergence Definition for Finite Difference Scheme. The definition is given by Convergence: for one-step schemes approximating a IBVP to be convergent we compare ...
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0answers
9 views

How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
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0answers
30 views

Show $A:= -\Delta^2 - \beta\Delta$ is sectorial

Let $U$ be a open, bounded with boudary sufficiently regular domain. I want to show that the operator $$ A:= -\Delta^2 - \beta\Delta $$ defined on $U$, for some $\beta\in\mathbb{R}$ and ...
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0answers
19 views

monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
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1answer
28 views

Euler equations

What's the relationship between the incompressible, free surface euler equations and the euler equations? Are the latter just the former when the free surface is identically zero?
7
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1answer
142 views

My first partial differential equation attempt

have I solved this correctly? My textbook is asking for the relation between $ \alpha $ and $ \beta $: $$ \frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}} $$ Textbook's proposed ...
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0answers
21 views

Local existence for semilinear wave equations

After having done some research, I could not find a reference for the following. Suppose I have a problem of the following type, on $(t,r) \in \mathbb{R} \times \mathbb{R}^2$: $$ \begin{array}{ll} ...
2
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1answer
55 views

Symmetry method for 2d heat equation

Suppose we have the pde $$ \frac{\partial p}{\partial t} = \frac{1}{4}\left( \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} \right) $$ Assuming a solution of the form, $$ ...
7
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1answer
267 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
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0answers
17 views

Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} ...
2
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2answers
45 views

Strong solutions to an elliptic PDE

I would like a reference for the following result (you can assume more regularity and replace $C^2(\bar\Omega)$ with $C^2(\mathbb R^n)$ if needed): Let $\Omega\subset\mathbb R^n$ be a bounded ...
3
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1answer
23 views

Defining a distribution

We fix the space $\mathcal{D}=\mathcal{C}^\infty_0(\mathbb{R}^n)$ as space of testfunctions. Let $(f_n)$ be a sequence of distributions with $\lim_{n\to\infty} f_n(\varphi)$ existing for all ...
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2answers
281 views

Time Derivative of PDE solution

Consider the PDE $$ \frac{\partial \psi}{\partial t}(t,x)=\psi''(t,x)+2\psi\psi'(t,x)-\tilde V'(x)~ on~ [0,D/2]\times(0,\infty)\\ \psi(0,t)=0, \psi(D/2,t)=-k, \\\psi(\cdot,0)=\psi_0 $$ for some ...
2
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1answer
36 views

Intuition to solving partial differential equations

I do not understand how to solve PDEs using the geometric method. I just do not understand the logic behind the solution. For example, the constant coefficient equation $$au_x + bu_y = 0,$$ where a ...
2
votes
1answer
49 views

Partial Differential Equation $\frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]$

In my research I have come across the partial differential equation \begin{equation} \frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]. \end{equation} ...
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0answers
32 views

How to compute an integral involving the error function

Solving a problem with one dimensional diffusion the following identity naturally arises $$\int _{-\infty }^{\infty }\frac{q}{2}{{\rm e}^{\eta\,t{\alpha}^{2}-\alpha \,x}} \left( {{\rm ...
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1answer
22 views

Uniqueness of solutions to the wave equation

we are given the problem $u_{tt}-c^2\Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $x\in\mathbb{R}^n$ and $u_0,u_1\in\mathcal{C}^1$ having compact supports. If a solution ...
2
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1answer
26 views

Heat Equation, possible solutions

NOTE: This is a homework problem. Please do not solve. I was given a problem that asked me to find a function of the form $u_n(x,t)=\chi_n(x) \cdot T_n(t) $ that solves the heat equation with the ...
0
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1answer
37 views

PDE - how to transform this pde to an easier one using change of varriable

I have this PDE its actually an ADE and I can put it in one of these forms. All characters in all four equations are constants with exception of x, y, t and C. I have listed the equations in order ...
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0answers
32 views

Boundedness of Helmholtz projection

Let's cosider the Helmoltz projection $P$ into solenoidal subspace defined as $$P=I-\nabla div\Delta^{-1}$$ What can I say about its boundedness as an operator in $H^s(\mathbb{R^3})$?
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15 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
3
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0answers
56 views
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mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
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0answers
22 views

Regularity for a vector transport equation

Let $\Omega\subset\mathbb{R}^N, N=2$ or $N=3$, be a smooth bounded domain and $T>0$. Given a vector function $h: \Omega\times (0,T)\to \mathbb{R}^N$ such that $\nabla\cdot v=0$. For any function, ...
2
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1answer
45 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
2
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0answers
39 views

Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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0answers
64 views

How could I solve this PDE?

Can anybody please let me know an idea about solving the following PDE for a given initial condition $T_0=0$ and boundary conditions $T(0,w)=u_0$? \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . ...
3
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1answer
78 views

$-\Delta u - \alpha u^{1/3} = 0$ implies $u \equiv 0$ if $\alpha$ is small

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ with smooth boundary. Let $u(x)$ be a $H^{1}(\Omega)$ solution of the equation $-\Delta u - \alpha u^{1/3} = 0$, $u|_{\partial \Omega} = 0$. The problem I ...
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0answers
54 views

Solution of Laplace equation for a regular hexagon

I am trying to analytically solve Laplace equation in a regular hexagon. My equation is $\nabla^2 \phi=0$ and boundary conditions are : $\phi = 0$ (at the base of hexagon) $\phi= \phi_1$ $\phi= ...
2
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0answers
67 views

Heat equation on a graph Laplacian

I would like to start with considering the time-dependent heat equation on a connected graph. To start, I will need to model it respect to time discretization. I mean I have to write something like: ...
0
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1answer
41 views

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $a > 0$ and $x(0) = 0.$

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $a > 0$ and $x(0) = 0.$ Let $f(x) = X'$ I, then, take the derivative of $f(x)$ which gives me $f'(x) = ...
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0answers
45 views

Leibnitz rule for fractional derivatives

I need to estimate the following norm $$\Vert fg\Vert_{H^{\frac{1}{2}}(\mathbb{R}^3)}$$ Is there some product rule for the fractional derivative?