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

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Derivation of the advection equation

Is there a good derivation of the advection equation available online? By that I mean the equation $\partial_t u = -\nabla( \vec{v} u)$ I know a good explanation for the one-dimensional case ...
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1answer
48 views

derivation of heat equation

In deriving the heat equation in the book it says Fourier's law says that heat flows from hot to cold proportionately to the temperature gradient. Therefore change of heat energy in $D$ is also ...
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1answer
45 views

Estimative with kernels of the Riesz transform

Let $x=(x_1,x_2)$ and $k(x)=\frac{x_1}{|x|^3}$, $|x|>0$. Consider $x,y\in\mathbb{R^2},\xi=|x-y|, \tilde{x}=\frac{x+y}{2}$. Then $|k(x-t)-k(y-t)|\leqslant C\frac{|x-y|}{|\tilde{x}-t|^3}$ when ...
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34 views

Show that $\lim_{a \to \infty} \sup_{n} \int_0^{T-a}||v_{n,r}(t+a)-v_{n,r}(t)||_{\mathbb{L}^2(\Omega_{2r})}^2dt=0.$

Let $\ 0 \leq t \leq t+a \leq T$, with $$\lim_{a \to 0} \sup_{n} \int_0^{T-a}\left\|u_n(t+a)-u_n(t)\right\|_{\mathbb{L}^2(\Omega_{r})}^2dt=0,$$ where $\Omega_r=\Omega \cap \left\{x \in \mathbb{R}^2; ...
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39 views

in PDE, $C^\infty_0$ and boundary value.

When I read PDE books as a novice, $\phi \in C^\infty_0(\Omega)$ often appears. Accidentally, given that condition I saw that function values in $\partial\Omega$ are zero. For example, a note from ...
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76 views

variational derivative

Let $\Omega \subset \mathbb{R}^n,\ n=1,2 \mbox{ or } 3$. Define the following energy $$E=\int_{\Omega} \frac{1}{\varepsilon}\left[f(u)+\frac{\varepsilon^2}{2}|\gamma(n)\nabla u|^2\right]\,dx$$ ...
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1answer
31 views

Uniqueness of a solution to a 1st order PDE

I'm asked to discuss the uniqueness of the solution for $$u_y+u_x=u-x-y$$ $$u(x,-2)=x$$ I've found the solution to be $$u(x,y)=x+y+2$$ But I don't know how to prove the uniqueness of the solution.
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235 views

Poisson Kernel for the half space and ball

I have 2 questions from Lawrence Evans' PDE book (pages 36 and 39 in the 2nd edition copy). The first is for green's function for a half space ...
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55 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
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1answer
41 views

Application of maximum principle

If I have a maximum principle of the form: If $\psi\geq 0$ of the boundary. Then $L\psi\geq 0$ on the domain $D$ implies $\psi\geq 0$ on the closure of $D$, for some linear operator $L$ defined in ...
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1answer
53 views

Elliptic partial differential equations

Consider the following elliptic PDE: $$ \Delta u=f(u), $$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have ...
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1answer
200 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
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1answer
116 views

Solution to a Partial Differential Equation

May I know how to solve the following partial differential equation? $\displaystyle\frac{\partial z}{\partial x}=A\displaystyle\frac{\partial z}{\partial y}+B\displaystyle\frac{\partial^2 z}{\partial ...
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1answer
61 views

Are pseudo(micro)-local operators pseudodifferential?

$\DeclareMathOperator{supp}{supp} \DeclareMathOperator{sing}{sing}$Let $\Omega$ be a domain with compact closure in $\mathbb R^n$. Consider a linear operator $A \colon X \to X$ satisfying one of the ...
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1answer
36 views

Solving PDE only using method of characteristics

Solve $aU_x+bU_y+cU$=0 using characteristic method. I know how to solve this by change of coordinates as in this article. But without changing coordinates how to do it with the method of ...
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1answer
53 views

Where to specify boundary conditions

The problem asks me where I need to specify boundary values for the linear PDE problem: $u_t + xu_x + yu_y = 0$ on the domain $\Omega = x^2 + y^2 \le 1$. Using characteristics I get that $u(x,y,t) = ...
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1answer
43 views

A subharmonic function

When $u=u(x)>0$ is a smooth non-constant function in $\mathbb{R^n}$ and is subharmonic in $\mathbb{R^n}$, i.e. $u\geq0$ in $\mathbb{R^n}$, can we conclude that $u$ is unbounded in $\mathbb{R^n}$ ...
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0answers
52 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
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1answer
45 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
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1answer
44 views

Isomorphism between Sobolev space and L^p

Let $L_1$ be an elliptic PDE operator $L_1:W^{2,p}\rightarrow L^p$ and $L_2=e^fL_1$ where f is a bounded function. I proved $L_1$ is an isomorphism, can I conclude $L_2$ is an isomorphism?
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1answer
26 views

shrinking an open domain with a smooth boundary - what is the new boundary?

Suppose $D$ is an open, convex and bounded set in $R^3$ with a smooth boundary. I want to shrink $D$ a bit but preserve the same 'shape'. What I'd like to do is to take inward normals at each point of ...
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1answer
70 views

solve partial differential equation

$$y^2u_x + xu_y = \sin(u^2) \\ u(x,0)=x$$ I get the projected characteristic curve on xy plane easily. However, cannot get the other one. actually the problem is getting the value of $U_{xx}, ...
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3answers
219 views

Physical meaning of boundary conditions in the diffusion equation

I want to simulate the diffusion equation numerically. $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ With the boundary condition $\frac{\partial u}{\partial x} \bigg|_{x=R}=0 ...
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1answer
152 views

Having trouble integrating for use of energy method to prove uniqueness

We are given $u_{tt} - c^2u_{xx} + ru_t$. To prove only one solution exists, I am taking w = $u_1 - u_2$, assuming they are both solutions to the given wave equation. So: $u_{tt} - c^2u_{xx} + ...
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1answer
51 views

Coordinate method for PDE

Solving the PDE $au_x+bu_y+cu=0$ The PDE is transformed by the coordinate method via, $\begin{cases}x'=ax+by\\y'=bx-ay\\\end{cases}$. What I don't understand is how should I know I have to pick ...
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2answers
175 views

Weird Al Yankovic's Partial Differential Equation

In Weird Al Yankovic's music video "White and Nerdy" (1:20-1:36), there are flashes of a partial differential equation: $$\left(-\frac{h^2}{2\mu}\nabla^2 - \frac{e^2}{r}\right)\psi(r)=E\psi(r)$$ ...
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1answer
92 views

PDE Questions! General Solution of Wave-kind Equations

I encountered an difficult wave equation plus an extra term which I have no clues how to solve as the following: Find positive functions $f$ such that $\frac{1}{2}\frac{\partial^2 f(x,y)}{\partial ...
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1answer
65 views

minimizes the functional to solve a pde

I am trying to do this exercise: Let $\Omega$ a open bounded domain in $R^n$. Consider the Dirichlet problem $$ \left\{ \begin{array}{ccccccc} -\Delta u = \lambda \sin (u) + f , \ \text{in ...
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1answer
33 views

Weak convergence of the 4-th degree of a weak convergent sequence

Good day! We solve an optimal control problem $$ J(u) = \|y - y_d\|^2 \to \inf $$ where $y$ is a solution of the PDE $$ \frac{dy}{dt} + Ay = Bu. $$ $A$ is a nonlinear operator, $(Bu, v) = ...
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0answers
41 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
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1answer
73 views

Existence of positive solutions of a linear PDE on closed manifolds

I was wondering is there a sufficient condition (or sufficient and necessary condition) for the existence of positive solutions of the following linear PDE on a closed manifold $(M, g)$, ...
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114 views

Passing to the limit in a PDE; problem with subsequence (please check my answer)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...
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1answer
45 views

Show that this functional is coercive - variational methods

For $u \in H^{1}_0({\Omega})$ ($\Omega$ is a domain open and bounded in $R^n$). Let $0 < \lambda < \lambda_1$ ($\lambda_1$ is the first eigen value of the laplacean) and a fixed $f \in ...
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2answers
91 views

Elliptic PDE, uniqueness of solution

I'm considering a partial differential equation of the form $$\nabla^2 u + \mathbf{a}\cdot\nabla u = 0$$ with Dirichlet boundary conditions, where $\mathbf{a}$ is a (smooth, nonconstant) vector ...
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51 views

Self-Adjointness of beam stiffness operator

I think it is well known that the operator $\frac{EI}{\rho} \frac{\partial^4}{\partial x^4}$ which arises from a standard Euler-Bernoulli beam is self-adjoint in $H$, where $H = L^{2}$, given ...
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1answer
129 views

Forced wave equation question?

I'm studying for my PDEs midterm and trying to do practice problems. I'm really not sure how to do this question - I've never seen anything like it. Thanks in advance for your help. Solve the ...
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49 views

How to solve this second order PDE

I have this differential equation. $$Dc''+H=0 $$ where the partial of the concentration is with respect to z the distance. H is the rate per unit volume of particles generated and D is the diffusion ...
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27 views

About an inequality equivalence in Optimal system

currently I'm working on an optimal control problem and I have what I think is a really simple question about it, The variable that I'm looking to optimize is $\lambda$ and it lies in an $X^d$ space ...
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1answer
218 views

Solve as a series the equation $u_t = u_{xx}, u_x {(0,t)}=0 , u{(1,t)} =1, u{(x,0)} = x^2$.

This question is from section 5.6 of Partial Differential Equations: An Introduction 2nd Edition by Walter Strauss 2008. I have approached this question by using the separation of variables $u(x,t) = ...
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1answer
54 views

Advanced Topic in Numerical solution of Differential Equations?

i) IF $\frac{dy}{dt} = - \frac{∂H}{∂z}, \frac{dz}{dt}= \frac{∂H}{∂y}$ where H is a function of $y$ and $z$, show that $H(y,z)$ is constant in time. ii) Take a $H(y,z) =Ay^2 + 2Hyz + Bz^2$ where ...
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Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
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1answer
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change of variable in mollifier

link : wikipedia Consider the following standard mollifier. $$\eta(x) = \frac{1}{z}\begin{cases} e^{-\frac{1}{1-\|x\|^2}}& \text{ if } \|x\| < 1\\ 0& \text{ if } ...
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1answer
38 views

Approximation of continuous functions by a functions with vanishing second derivative

Denote by $C^n[-\infty,+\infty]$ the class of functions which: have finite limits at $\pm \infty$; and are differentiable $n$ times on the line, with all these derivatives bounded. Denote by $C^3_0$ ...
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1answer
213 views

Hadamard variational formula Evans chapter 6 problem 15

This is Evans' chapter 6 problem 15. Consider a family of smooth, bounded domains $U(\tau) \subset \mathbb{R}^{n}$ that depend smoothly upon the parameter $\tau \in \mathbb{R}$. As $\tau$ changes, ...
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2answers
90 views

mathematical biology (steady-states)

non-dimensionalisation equation: \begin{equation} \frac {du}{d\tau}=\frac{\overline{\lambda}_{1} u}{u+1} -\overline{r}_{ab}uv -\overline{d}u \end{equation} where $\overline{\lambda}_{1}= \frac ...
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1answer
51 views

Sobolev Inequality with powers

The following is one of the Sobolev Inequalities as stated in Gilbarg & Trudinger's text on elliptic PDEs: $\displaystyle W_{0}^{1,p}(\Omega)\subset L^{\frac{np}{(n-p)}}(\Omega)$ for $p<n$. ...
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38 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
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1answer
57 views

Does $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ give something useful?

If $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ is there any way to extract a strongly convergent subsequence of $u_n$ in some useful space for PDEs? Or ...
3
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2answers
190 views

How find this $\frac{yf_{y}-z}{f_{x}}+\frac{xf_{x}-z}{f_{y}}-xf_{x}-yf_{y}+x+y+z=C$ solution

In plane $R^3$,Find $z=f(x,y)$, such the length of the portion of any tangent line to the astroid $$z=f(x,y)$$ cut off by the coordinate axes is constant $C$, This problem is from this post (when I ...
4
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1answer
71 views

How find a solution to this PDE $\frac{xf'_{x}}{f'_{y}}+\frac{yf'_{y}}{f'_{x}}+x+y=C$

let $C$ is give the constant ,if the function $f(x,y)$ such $$\dfrac{xf'_{x}}{f'_{y}}+\dfrac{yf'_{y}}{f'_{x}}+x+y=C$$ Find the all $f(x,y)$ I found this problem one solution: ...