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

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7
votes
1answer
478 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
2
votes
1answer
37 views

The conservation of a critical non-linear dispersion equation.

Consider the non-linear problem $$ \frac{1}{i}\frac{\partial{u}}{\partial{t}}-\frac{d^2u}{dx^2}=\sigma|u|^{\lambda-1}u$$ $$u(x.0)=f(x)$$ Suppose that $u$ is a smooth solution that decays ...
2
votes
1answer
174 views

Heat equation with problematic boundary conditions

I'm asked to solve the heat equation $$u_t = \kappa u_{xx}$$ for $t \ge 0$, $0 \le x \le L$, given boundary conditions $$u_x(0,t) = u_x(L,t) = 0$$ and an initial condition $$u(x,0) = f(x) = ...
0
votes
2answers
113 views

General Technique of Method of Characteristics for PDE

I'm working problem 2.5.1 in Evans PDE book. The equation is $u_{t} + b*Du + cu = 0$ with the initial condition $u(x,o) = g(x)$. I fix x,t and define. $z(s) = u(x + sb,t + s)$ so $z'(s) = ...
3
votes
1answer
326 views

Boltzmann's transformation and change of variables.

Boltzmann’s Transformation is used (among other things) to convert Fick's second law into an easily solvable ordinary differential equation. It uses the variable $\xi=\frac{x}{2 \sqrt{t}}$. As far ...
4
votes
0answers
189 views

Existence of weak solution in Sobolev space $W^{1,p}$

Let $B\colon W^{1,p}\times W^{1,q}\rightarrow \mathbb{R}$ and $\frac{1}{p}+\frac{1}{q}=1$ , $p>2$ where $$B[u,v]=\int \left( Du \,Dv+ h u v \right)\, dx$$ where $h>0$. I am interested to prove ...
3
votes
0answers
678 views

Help with a non-linear partial differential equation

I am wondering whether someone can help me with a non-linear PDE: $$\frac{\partial^2\phi}{\partial t^2} = c\frac{\partial^2\phi}{\partial x^2} \left(\frac{\partial\phi}{\partial x} \right)^{n-1}$$ ...
3
votes
0answers
199 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
2
votes
0answers
61 views

Solving an eigenvalue problem on the open unit rectangle

Let $\Omega=(0,1)\times(0,1)$ and consider the boundary value problem $$\begin{cases}\Delta^2u=f\\ u(x,y)=\Delta u(x,y)=0,& x,y\in\partial\Omega \end{cases}$$ I want to solve this boundary value ...
5
votes
1answer
91 views

Reference for compution adjoint of the operator $L=\Delta^2$

I need to use adjoint operator of the partial differential operator $L=\Delta^2$, where $\Delta$ denotes the Laplacian. I do not want to put this computation in my thesis, because I feel is a bit ...
1
vote
1answer
75 views

How to show that $\int_0^T (u_m(t), u_m(t))_{H} \to \int_0^T (u(t), u(t))?$ (Bochner space)

Let $p \geq 1.$ Let $$W = \{ u \in L^p(0,T;V) : u' \in L^{q}(0,T;V^*)\}$$ where $V \subset H \subset V$ is a Gelfand triple ($V$ Banach and $H$ Hilbert), and $p$ and $q$ are conjugate indices. The ...
4
votes
3answers
375 views

How to solve vector-valued first order linear pde?

Is there an analytical solution to the pde system? $$\frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} = 0$$ $$\frac{\partial f}{\partial y} - \frac{\partial g}{\partial x} = 0$$ More ...
3
votes
0answers
79 views

Trace spaces of Orlicz-Sobolev spaces.

Recently I have the need of study trace spaces of Orlicz-Sobolev spaces. By looking in google I have discovered in this PDF page 14 (not only in the PDF), that the main contributions come from the ...
1
vote
0answers
45 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
2
votes
1answer
92 views

How to prove that the circumference is $C^k$?

Notation: $U$ is a open subset of $\mathbb{R}^n$; $\partial U$ is a boundary of $U$; A $C^k$ function is a function $k$-times continuosly differentiable; $B(x_0,r)$ is a ball in $\mathbb{R}^n$ with ...
2
votes
0answers
59 views

Extending $W^{2,n}(\Omega)\cap C^2(\Omega)$-functions to $W^{2,n}(\Omega)$

I have the following theorem: Let $u\in C^2(\Omega)\cap W^{2,n}(\Omega)$ satisfy $Lu\geq f$ for $f\in L^n(\Omega)$. Then for any ball $B_{2R} = B_{2R}(y)\subset\Omega$ and $p>0$ where $u$ is ...
2
votes
0answers
67 views

On solving heat equation from ancient time

The problem asks to find all $C^2$ solution for $$u_t-u_{xx}=t-x^2,\quad (t,x)\in \mathbb{R}^2$$ satisfying $$\lim_{|x|+|t|\to \infty}\frac{|u(x,t)|}{|x|^5+|t|^5}=0$$ The typical heat equation with ...
5
votes
1answer
425 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...
4
votes
1answer
443 views

PDE - solution with power series

I am learning this method for solving PDSs by means of power series. Since I am studying it with lecture notes and I can't find any other book that describes this method, I am going to summarize it ...
2
votes
0answers
38 views

eigen problem for direct scattering method

Consider the KdV equation $$u_{t}+6uu_{x}+u_{xxx}=0$$ with initial condition $$u(x,0)= \begin{cases} 1 &\text{if } x \in [-1,0] ,\\ 0 &\text {if } x \in ...
2
votes
1answer
678 views

PDE- Wave equation on semi-infinite string

I am becoming frustrated in trying to understand the wave equation in the semiinfinte case: $ u_{tt} -c^2 u_{xx} =0 $ when $ x\geq 0 $ $u(x,0)=f(x) $ $ u_t(x,0)= g(x) $ and $ u(0,t)=0 $ or $ ...
0
votes
1answer
223 views

PDE continuous dependence on data definition

Let us have a linear partial differential equation $$u_t + Au = f.$$ If I know that $$|u| \leq C(|u_0| + |f|)$$ then does that mean I know "continuous dependence on initial data"? Is this common in ...
4
votes
2answers
174 views

Why do all solutions to this equation have the same form?

In this paper on page 45, the authors state that Let's assume we know that $$ w \times dw = d\varphi + \sum_{j=1}^3\alpha_jdx_j, \tag{1}$$ where $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, ...
3
votes
2answers
94 views

How to integrate this differential form on the boundary of the cube

The setup. Assume $u = u_1+iu_2: \mathbb{R}^3 \to \mathbb{C}$ and we have the differential 1-forms $$ \star\xi=-x_2 dx_3 + x_3 dx_2 $$ and $$ u \times du = \sum_{i=1}^3 (u \times \partial_i u) dx_i = ...
1
vote
3answers
103 views

Is there a lower bound for $\int_{B_{r}}f$ when $f$ is a positive function?

Let $f$ be a positive Lebesgue integrable function on a ball $B_{r}\in\mathbb{R}^n$. I'm looking for a positive constant $c$ depending only on $f$ that satisfies $$c\omega_nr^n\leq\int_{B_r}f$$ where ...
1
vote
0answers
193 views

Separation of variables - Facing difficulty in application of Orthogonality Condition

I am trying to model transient cooling of two concentric cylinders sharing an interface along the length. Heat will flow in radial direction only. Radius of inner cylinder (or inner radius of outer ...
0
votes
1answer
179 views

How to write the expression with multi-index notation?

I need some help for writing the following expression with multi-index notation, $$\sum_{i_1, \ldots, i_p=1}^n \frac{\partial^{2p}}{\partial x_{i_1}^2\ldots \partial x_{i_p}^2}f(x, \xi),$$ where ...
9
votes
2answers
247 views

Do we need to identify dual spaces in PDEs?

In PDEs we often use the fact that we can identify dual spaces eg. $L^2(0,T;V)^* = L^2(0,T;V^*)$ in the sense that $$u_t + Au = f$$ where $u_t$, $f \in L^2(0,T;V^*)$ and $A:L^2(0,T;V) \to ...
3
votes
1answer
130 views

Schwartz kernel theorem for induced distributions…

I'm studying periodic pseudo-differential operators on torus and I have a question concearning the Schwartz kernel theorem: If $A:C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n)$ is a ...
1
vote
1answer
75 views

To show that $ J(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1} $ is not bounded above for $1 < p < 2^*-1$

For a bounded $ \Omega\subset\mathbb{R}^n $ with smooth boundary, and for $ 1 < p < \frac{n+2}{n-2} = 2^* -1 $ where $ \frac{1}{2^*} = \frac{1}{2}-\frac{1}{n}$, I have the functional $ J : ...
2
votes
1answer
62 views

Antiderivative of a function arised in KdV equation

I am computing the third non-trivial conservation law of KdV equation $$u_{x}+6uu_{x}+u_{xxx}=0$$ via the power series expansion method (Here we consider real-valued solutions only). I was given an ...
8
votes
1answer
379 views

Is this bootstrap argument correct?

The setup. Assume $\Omega \subset \mathbb{R}^3$ bounded and bilipschitz equivalent to the unit cube and has smooth boundary. Let $v \in W^{1,2}(\Omega,\mathbb{C})$ be a weak solution of $$ ...
7
votes
2answers
199 views

Definition of weak solutions from geometrical point of view

Why are weak solutions defined like: A function $u \in H^1(\Omega)$ is a weak solution of $$ Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega $$ if ...
2
votes
2answers
589 views

strong maximum principle - harmonic function

Consider the following the theorem in the classical PDE book of Evans( chapter 2 ) : (part of the strong maximum principle) Let $U$ a open set in $R^n$ and $u \in C^2 (U) \cap C(\overline{U})$, with ...
3
votes
2answers
776 views

Geometric proof of rotational invariance of Laplacian

I am just looking for a more intuitive proof that the $\nabla^2$ operator is rotationally invariant in Euclidean space. That is: If $u(x)$ solves $\nabla ^2 u=0$, then $v(x)=u(Rx)$ solves it too, ...
3
votes
0answers
56 views

Ergodic mean for Schrodinger flow

Let us consider the linear Schrodinger equation in $\mathbb{R}^N$ $$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$ with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ its solution. ...
3
votes
1answer
133 views

Elliptic theory (existence and regularity) for pde of complex functions

I'm aware that this is a rather general question, but I only need some hint to literature. The setup. I'm studying the existence and regularity of weak solutions to linear elliptic pde of the form $$ ...
2
votes
1answer
36 views

Can one use the IFT on Banach spaces and the simple harmonic oscillator to say that there is a solution for the motion of a pendulum?

Let $W^{k,p} (S^1)$ be the Sobolev space of $k$ times weakly differentiable periodic functions, all whose weak derivatives upto $k$ are in $L^p(S^1)$. Consider the non linear operator $A_t : W^{k,p} ...
7
votes
1answer
275 views

Characteristics in the theory of PDE's - What's Going On?

What's really going on as regards characterstics in the theory of PDE's? (Hopefully you wont have to check the links provided below, included for those who are interested really) For second order ...
3
votes
1answer
66 views

Nonlinear parabolic PDEs, what methods/techniques for existence?

I am curious what kinds of techniques one uses to show existence of PDEs with nonlinearities. I am aware of: 1) Minimisation problems 2) Semigroup (both of which I'd like to avoid) For linear ...
2
votes
1answer
77 views

Why is the case $1/2<\rho\leq 1$ trivial in proving the following inequality?

I'm studying Elliptic Partial Differential Equations by Q. Han and F. Lin. In Lemma 1.41 is given the elliptic equation $D_j(a_{ij}D_i u)=0$ where the coefficient matrix $(a_{ij})$ is constant ...
2
votes
1answer
58 views

Why does the following inequality holds for any weak solution $u\in C^1(B_1)$ of uniformly elliptic equation $D_i(a_{ij}D_ju)=0$?

Now I'm studying "Elliptic Partial Differential Equations" by Q.Han and F. Lin. Throughout the section 5 of the chapter 1, $u\in C^1(B_1)$ is a weak solution of $$D_i(a_{ij}D_j u)=0$$ where ...
2
votes
0answers
195 views

Gagliardo Nirenberg Sobolev inequality

Assume that $f$ satisfies the equality in the Gagliardo Nirenberg Sobolev inequality for the best constant. What can be said about $f$?
1
vote
1answer
234 views

Steklov Average in time and spatial gradient can be interchanged?

I'm trying to understand a proof on Steklov average and weak derivatives. Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $T>0$, $h\neq 0$ and $u\in L^2(0,T;H^1(\Omega))$ and extend $u$ by ...
1
vote
2answers
202 views

How exactly does the constant $C$ in the Sobolev inequality depend on the domain?

The Sobolev inequality theorem -as stated here- says Let $U$ be a bounded open subset of $\mathbb{R}^N$, with a $C^1$ boundary. Assume $u \in W^{k,p}(U)$. If $k<n/p$ then $u \in L^q(U)$, where ...
2
votes
0answers
270 views

Gradient of an harmonic function

Let $ M $ be a Riemannian manifold and let $ f $ be an harmonic function on $ M $. By Unique continuation theorem we can assert that if $ \nabla f = 0 $ on an open subset $ \Omega \subset M $ then ...
2
votes
1answer
180 views

theorem 1 chapter 2 - Evans PDE

My doubt is about the proof of the theorem 1 section 2.2.1 of the evans pde classic book. My doubt: Consider the function $$\Phi(x) = \begin{cases} - \frac{1}{2 \pi} \log |x| & \text{if $n= ...
4
votes
2answers
78 views

Finding the most general class of solutions to $x\partial_{y}f = y\partial_{x}f$

Consider the following PDE for $f(x, y)$: $$ x\frac{\partial f}{\partial y} = y\frac{\partial f}{\partial x}\tag{1} $$ Clearly, one can separate the variables, so take $f(x, y) = p(x)q(y)$: $$ ...
3
votes
1answer
68 views

Numerical solution to diffusion-like equation with changing sign

I am trying to numerically solve an initial value problem $$ \frac{\partial f}{\partial t} = \frac{1}{x} \frac{\partial^2 f}{\partial x^2}$$ where $f = f(x,t) \text{ for } x \in [-1,-1],\ t \in [0, ...
2
votes
0answers
80 views

Solving PDE equation

My main problem to solve is anisotropic wood (2D) with tree rings paralel and all in one direction. I'm having problem solving the equation $$au_{xx} + bu_{yy} = -c.$$ It's a non time dependent ...