Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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2
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1answer
343 views

The PDE $u_t=u_{xx}-u_{yy}$: The simplest linear second-order PDE that isn't elliptic, parabolic, or hyperbolic.

I know that there have been several questions on here and stackexchange about linear PDE's which don't fall into the standard classification, but I had a more focused question which I haven't seen ...
3
votes
0answers
148 views

Exponential decay for the gradient of a solution

Dear all, I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$: $$ \sqrt{-\Delta+m^2} u +u= f(u), $$ where I can assume that $m \neq ...
15
votes
1answer
500 views

The Riemann zeros and the heat equation

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as $$ \Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du, $$ where $\Phi(u)$ is defined as $$ ...
8
votes
2answers
737 views

PDE with the Jacobian Determinant

Hello, Could you please help me in answering the following question? Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not ...
3
votes
1answer
234 views

Hormander's bracket condition for the adjoint of an operator

Let $X_0, X_1, \dots, X_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator $$ L = \sum_{i=1}^k X_i^2 + X_0~. $$ Here, I assume that Hörmander's bracket condition is ...
1
vote
1answer
251 views

Please recommend some classical books or articles on the compressible Euler equations !

Please recommend some classical books or articles on the local well-posedness result of compressible Euler equations ! The main aim is that I want to learn some basic methods and techniques about the ...
3
votes
1answer
186 views

C^\infty versus semiclassical wavefront sets

Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ ...
5
votes
0answers
95 views

Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ ...
0
votes
0answers
115 views

damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation. My question is...why is the ...
2
votes
1answer
135 views

Distributional limits concerning the regularity of Maxwells equations

This question is related to my previous question about the regularity of the Maxwell equations. Assume we are working on a space where there are only electric point charges, $(q_i)$, and a blob of ...
74
votes
16answers
5k views

Does Physics need non-analytic smooth functions?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ...
5
votes
3answers
391 views

Divergence form Elliptic PDE Removable Singularity/Regularity Question

Idea Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...
5
votes
1answer
350 views

propagation of singularities & the Schrodinger equation

I've been thinking about the following propagation of singularities result: Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose principal symbol ...
6
votes
1answer
661 views

fixed point arguments in PDE

I was curious whether anyone knows of some examples in PDE where a standard fixed point argument fails to show the existence of a solution but one can apply one of the more advanced fixed point ...
1
vote
1answer
240 views

Does this PDE has a general solution? [closed]

$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$ K and u are constants. If no,how to get a asymptotic solution?ie,linearize
0
votes
1answer
409 views

Showing a coercivity condition for this bilinear form

Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to ...
3
votes
3answers
277 views

Is there a good metric under which a sequence of compact sets can converge to an infinite dimensional set?

I have a sequence of finite-dimensional, compact sets in $L^2(\Omega)$, where $\Omega\subset \mathbf{R}^2$ is closed and bounded. The dimension grows monotonically with the sequence, and there is no ...
1
vote
2answers
201 views

Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definition regarding the following: Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
9
votes
1answer
621 views

Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...
4
votes
1answer
525 views

Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello! Let's say I have $(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$ with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ...
8
votes
1answer
491 views

The Speed of Gravitational Waves in General Relativity

Is it possible to mathematically prove that the speed of gravitational waves in general relativity equals the speed of light, without linearizing the Einstein Field Equations? The approach via the ...
1
vote
1answer
65 views

Finding singular “solutions” to the Dirichlet problem for Schrödinger operators that do not admit smooth solutions

Suppose that $\Omega\subset \mathbb{R}^n$ is a smooth open region and that $V:\Omega\to \mathbb{R}^+$ be a positive smooth function. Then we have a family of operators $$L_\epsilon =-\Delta -\epsilon ...
2
votes
1answer
277 views

Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$: $$ \int_{\Omega} \nabla u \nabla v d x = ...
1
vote
0answers
149 views

two polynomial equations

Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system $$f_{11}+2tf_{12}+t^2f_{22}=0$$ $$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$ ...
3
votes
1answer
152 views

Subharmonic envelope

I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ...
6
votes
1answer
314 views

Green functions on Riemann surfaces

Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ ...
1
vote
1answer
155 views

Well-posedness of Euler-Poisson system for semiconductors

Is there anyone can recommend me some important literature references about the well-posedness of both Cauchy problem and initial boundary value problem of Euler-Poisson system for semiconductors? ...
0
votes
0answers
167 views

hitting probability for integrated Ornstein-Uhlenbeck process

Consider an Ornstein-Uhlenbeck position process: $dV_t=dB_t-\lambda V_tdt$ $dX_t=V_tdt$ where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ . Let $r>0$ and $S_r$ be the ...
2
votes
2answers
409 views

Easy question on Sobolev spaces

I understand that this question would be trivial for experts, sorry for that, I just need to clarify things. So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W_p$, $W_q$ are ...
2
votes
1answer
299 views

Weak divergence implies weak differentiability of components?

Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$. We say that $\sigma$ has weak divergence if there exists ...
1
vote
1answer
162 views

A Cauchy problem for an iterated Euler-Poisson-Darboux eqaution

Good morning, I'm interested in solving a Cauchy problem for the iterated singular EPD. Well, Weinstein (On a class of PDEs of even order, 1955) showed how the decomposition formula leads to the ...
0
votes
0answers
81 views

About the boundedness of the derivative of a function which is in a special function space.

If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, $f (t,x) : bounded\; on \; [0,T] \times \Bbb R^n $ then how can I conclude that $$ \left \| \frac{\partial f}{\partial t} \right ...
6
votes
0answers
273 views

Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by $$\frac{\partial}{\partial t} ...
4
votes
2answers
244 views

Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform

Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose $$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$ Is $a(x,0)=0$, $\forall x\in[0,1]$? ...
2
votes
1answer
149 views

Reference request: Stability / instability theory for periodic orbits of partial differential equations

I am looking for references regarding the stability / instability of a periodic solution to a partial differential equation / evolution equation in infinite dimensions. Suppose we have a periodic ...
8
votes
2answers
1k views

what's the idea behind Carleman estimate

A standard Carleman-type estimate is of the form $$ \sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty} $$ ...
1
vote
2answers
317 views

Variational problems whose lagrangian density depends on derivatives higher than 1.

The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
0
votes
1answer
542 views

The conormal derivative of a function

Hi! I was wondering about the definition of the conormal derivative of a function $u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered as functionals on ...
1
vote
1answer
348 views

Elliptic Differential Equations with rough boundary data

Question stated roughly: Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one ...
4
votes
1answer
133 views

Interpretation of a parameter in forming a pseudodifferential operator

In Zworski's Semiclassical Analysis, he defines the following method of quantization: for a symbol $a = a(x,\xi) \in \mathscr{S}(\mathbb{R}^{2n})$ and $u \in \mathscr{S}(\mathbb{R}^n)$, $$ ...
2
votes
1answer
200 views

Solvability of a nonlinear elliptic equation

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation \begin{equation} -\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0. ...
2
votes
3answers
875 views

Jacobi method on first order partial differential equations

Hi, I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject. Thanks in advance
3
votes
1answer
187 views

Reference Request: Schauder theory for fourth-order parabolic equations

I am looking for a treatment of fourth order parabolic equations in Holder spaces. More precisely fourth order analogues of Theorems 5.1, 5.2, and 10.1 in Chapter IV of Linear and quasilinear ...
5
votes
3answers
798 views

Reference request: parabolic PDE

I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics. I think I have a firm grip ...
6
votes
4answers
616 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be smooth compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by \begin{equation} ...
2
votes
1answer
298 views

Nash inequality on a compact domain?

I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the ...
4
votes
1answer
590 views

Solving PDE with Cauchy - Kowalewski Theorem

Hallo, I have the following PDE that I am trying to solve via the Cauchy-Kowalewski Theorem. But I have no idea how to do it or if its possible. Maybe one of you has an idea. Here is the problem: Let ...
2
votes
1answer
206 views

A question on Schwartz distributions

I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...
0
votes
1answer
144 views

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is : ...
4
votes
3answers
881 views

Quasi-linear System of First Order P.D.E.s of “Mixed” type

In my research work, I am dealing with a quasi-linear system of first order p.d.e.'s with two independent variables (say $x_1$ and $x_2$) and four dependent variables (say $u_1(x_1,x_2)$, ...