0
votes
0answers
9 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
2
votes
0answers
20 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
0
votes
1answer
18 views

What are good resources for learning Numerical methods for Partial Differential Equations?

I'm having an undergraduate course on Numerical Solutions to Ordinary and Partial Differential Equations. I need online resources to supplement my study preferably videos and books. I want to build a ...
0
votes
0answers
20 views

Elliptic regularity of Dirichlet problem

Suppose $\Omega\subset\mathbb{C}$ is a simply connected domain with $C^\infty$ boundary. Consider the following Dirichlet problem $$\Delta u |_{\Omega}=0 $$ $$u|_{\partial\Omega}=f$$ Under what ...
1
vote
0answers
43 views

Applications of PDEs

I teach an undergrad ODE course. As I have completed basically all the material, I thought it would be nice to give the students a brief introduction to PDEs. At the end of the lecture, I said that ...
1
vote
0answers
29 views

Regularity of Dirichlet Eigenvalues on Lipschitz Domain

What kind of regularity do we generally have for weak solutions to the Dirichlet problem? $$(\Delta+\lambda)u=0 \textrm{ in }U$$ $$u=0 \textrm{ on }\partial U $$ where $U$ is a planar domain with ...
1
vote
0answers
18 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
1
vote
0answers
26 views

Reference needed for the following sobolev inequalties

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
0
votes
0answers
14 views

Classification of second order PDE

I am trying to understand the classification of second order PDE's from this article. In page 45, line 1 can somebody please explain to me how $\partial^2U \over {\partial x_i\partial x_j} $ was ...
0
votes
0answers
16 views

Reference for elliptic regularity for $-\triangle \phi + u \cdot \nabla\phi=f$ under minimal assumptions

I have a distributional solution to $-\triangle \phi + u \cdot \nabla \phi= f$ in $U \subseteq \mathbb{R}^n$ and $\phi=0$ on $\partial U$. I have that $U$ is open, bounded, connected, ...
2
votes
1answer
19 views

how to prove that this weak solution is subharmonic?

My question is about this article http://hal.inria.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf. My question is : Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = ...
2
votes
0answers
30 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
0
votes
1answer
22 views

Reference needed for: $u \in H^1(0,T;L^2)$ if and only if $\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$

There is a result of the form: a function $u \in H^1(0,T;L^2)$ if and only if $$\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$$ holds for all $h \in [0,T]$. I have only seen one place ...
0
votes
0answers
57 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
1
vote
0answers
29 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
0
votes
1answer
28 views

Some extensions of Weyl's lemma

The Weyl's lemma said that: If $u$ is a continuous function on the open set $\Omega$ such that it satisfies $\Delta u=0$ in the distributive sense, that is $$\int_\Omega u \Delta\phi = 0$$ for all ...
1
vote
1answer
26 views

Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
0
votes
0answers
30 views

Reference request on Linear partial differential equations

I'm looking for recent references on linear partial differential equations, containing topics similar to L. Nirenberg's Lectures on Linear Partial differential equations, namely discussion on ...
0
votes
1answer
26 views

weak solution of Dirichlet problem in Lipschtiz domain with non zero boundary data

Let $\Omega$ a bounded and open with Lipschitz boundary. I know that exists the trace operator in the case of this $\Omega$. My question is : When $\Omega$ is bounded and open with Lipschitz ...
1
vote
0answers
19 views

Symmetrisation of system of conservation laws

Suppose we have a system $$\partial_t \rho +\nabla_x\cdot \upsilon(\rho)=0,$$ where $x\in \Bbb R^3$, $\rho:(t,x)\to \Bbb R^5$, $\upsilon: \Bbb R^5\to \Bbb R^3$. Suppose also that there exists a ...
4
votes
2answers
85 views

Solving wave equation

I want to solve the following PDE: $$\begin{align} u_{tt}&=c^2u_{xx}-\gamma u_x, \quad 0<x<1, \quad t>0,\\ \\ u(0,t)&=u(1,t)=0, \\ u(x,t=0)&=x(1-x),\\ u_t(x,t=0)&=0. ...
2
votes
1answer
37 views

Reference on $\mathcal{L}^p(I;X)$

I am doing some reading on evolution equations, and $\mathcal{L}^p$ spaces with functions with values in a Banach space $X$ appears rather often. However I have not found a comprehensive reference ...
2
votes
1answer
86 views

$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required

Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where ...
0
votes
1answer
37 views

Textbook for partial differential equations with the tools of complex analysis

Is there a textbook in partial differential equations that has a lot of solved excercises by using complex analysis tools?
1
vote
0answers
17 views

Characteristic Method for linear equation

I'm looking here for a reference to solve the following equation with the method of characteristic $$ \partial_t U + A\partial_x U = f$$ $$U(t=0,x)=U_0(x)$$ $$B(t,x=0)U = 0$$ where $(t,x)\in ...
7
votes
2answers
140 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
5
votes
2answers
97 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
0
votes
0answers
36 views

Reference needed for integration on boundary of Lipschitz domain

I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in ...
1
vote
2answers
39 views

Nodes of eigenfunctions and Courant's nodal domain theorem

I am looking for a reference for properties of eigenfunctions of the Laplacian (on the Euclidean plane, and maybe also Laplace-Beltrami on a general manifold): The discreteness of the set of ...
5
votes
1answer
140 views

Spherical means (Kirchoff's formula) for variable speed wave equation

Suppose $$ \begin{cases}u_{tt} - \Delta u = 0 \quad \textrm{ in } \mathbb R \times \mathbb R^n \\ u(0,x) = f(x); \quad u_t(0,x)= g(x). \end{cases} $$ then, depending on the dimension $n$, we have a ...
1
vote
0answers
18 views

Hyperbolic PDE with Flux Singularity

Consider the system of balance laws $$\partial_t u+\partial_x F_{\epsilon}(u) = S (u),$$ where $u = u(x,t) \in R^n$ denotes the state vector, $F: R^n \to R^n$ is the flux and $S:R^n \to R^n$ some ...
2
votes
2answers
90 views

The Poisson problem $-\Delta u =0$ with $u=g$ on the boundary where $g \in H^{\frac{1}{2}}$

Consider $$ \begin{align} -\Delta u =0 & \text{on $\Omega$} \\ u = g & \text{on $\partial\Omega$} \end{align} $$ where $g \in H^{\frac{1}{2}}(\partial\Omega)$. It seems there exists a ...
3
votes
0answers
29 views

Schauder estimate with right hand side in $L^n$.

The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have ...
1
vote
4answers
130 views

Pde book suggestion.

I am studying PDE. And I want to know introduction to PDE book's names, which contain direclet problem, Sturm liouville problems, cauchy problems, euler, eigen functions and like this. But the ...
1
vote
1answer
28 views

Fundamental solution and dissipation result for biharmonic Heat equation

I guess this is easy a very easy question for some people. References would be appreciate : What is the fondamental solution of the biharmonic heat equation ? and how fast is it decreasing in time ...
7
votes
2answers
159 views

Wave-Particle Duality in PDE?

I am reading Arnold's Lectures on Partial Differential Equations. It is definitely a good book, yet sometimes I am a little bit confused. One theme of the first chapter seems to be From the ...
0
votes
0answers
20 views

Elliptic equations in the Heisenberg group.

What books on elliptic equations in the Heisenberg group do you recommend? I am interested in studying on regularity problems for these equations. I have read some papers on the subject, but I seem ...
3
votes
0answers
25 views

Finite Difference Methods for arbitrary elliptic PDE

I am looking for textbook references that describe lattice numerical methods for arbitrary elliptic PDEs, particularly finite difference schemes and particularly in 2d. The few references that I have ...
2
votes
1answer
46 views

Are there any other materials on Hamilton-Jacobi equation besides the book by Evans?

I've been recently reading the book on PDE by Evans,and I had a hard time understanding the part about Hamilton-Jacobi equation in chapter 3.I wonder if there are any other materials on this ...
2
votes
1answer
32 views

Reference request: Energy of a superposition of waves

Suppose we have two waves $u_1$ and $u_2$ satisfying $$ \begin{cases} \partial_{tt}u_i = \Delta u_i\\ u_i(0,x) = f_i(x) \\ \partial_t u_i(0,x) = g_i(x) \end{cases} $$ in $\mathbb{R}^+ \times ...
2
votes
1answer
38 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
3
votes
1answer
105 views

What should everyone know about PDEs?

Suppose I'm going into a field where it's unlikely that I will ever see a PDE. Assuming I have a graduate level background in analysis, what should I know about PDEs to have a general understanding of ...
6
votes
6answers
344 views

Good PDE books for a graduate student?

I am now a graduate student in mathematics, and I really want to learn more about PDE. I would say I have a very solid foundation in soft analysis, including functional analysis and harmonic analysis, ...
9
votes
1answer
279 views

Effect of pullback of differential forms on an ideal

Say that the exterior differential system (EDS) corresponding to a PDE system is: $$df-f_x\,dx-f_y\,dy-f_w\,dw-f_z\,dz=0,\\ a_1\,f_x+a_2\,f_y=0,\tag{sys}$$ Of course we also require the independence ...
2
votes
0answers
22 views

Time continuity of a solution to the generalized porous medium equation

Let $\Phi(x,s)$ be a smooth, non-negative function with $\Phi(\cdot,0)=0$ and $\Phi$ increasing in $s$ and $\partial_s \Phi(x,0)=0$ and $\Omega$ be a bounded smooth domain. Now assume there is a weak ...
6
votes
0answers
61 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
1
vote
0answers
24 views

Find the limit of the PDE

Consider two following PDEs on functions $\psi_{1,2}(x,\lambda)$, $x \in \mathbb R^n$ depending on parameter $\lambda \in \mathbb C^n$: $$ \Delta_x \psi_1(x,\lambda) + 2i\lambda \cdot ...
0
votes
0answers
20 views

Establishing recurrence and positive recurrence of Markov processes via “barriers”?

I've been reading the book by Wentzell and Freidlin on dynamical systems with small random perturbations. On page 42 it's stated: It is possible to give stronger conditions for recurrence and ...
1
vote
2answers
69 views

Book searching in Elliptic Equation

I am learning a course with the subject of Elliptic Equations. If you know about it, please recommend me a book on Elliptic Equations. And if that's possible, someone post these books/author/...that ...
1
vote
1answer
139 views

Comprehensive references on partial differential equations

How do the three volumes by Taylor's "Partial differential equations" compare with the two volumes with the same title by Friedrich Sauvigny's as a reference for study? What are the good and bad ...