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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

2 votes
1 answer
55 views

Literature on Lateral cauchy problem

Can you please provide me with books that deal with lateral cauchy problem? Also introductory articles are fine by me. Is this topic covered in books like Evans', Taylor's, Hilbert-Courant's or othe …
Alan's user avatar
  • 1,594
1 vote
1 answer
251 views

A question on theorem 1.1 of Fritz John ultrahyperbolic pde

I have the following paper: Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), no. 2, pages 300-322 doi:10.1215/S0012-7094-38-00423-5 Now …
Alan's user avatar
  • 1,594
2 votes
1 answer
828 views

Lebesgue Riemann Theorem.

Does someone know where may I find the general proof of Riemann Lebesgue theorem which states that Let $1\leq p \leq \infty $ $M= \prod_{i=1}^{n} (a_i,b_i)$, and $u \in L^p$ , define $u_\nu = u(\nu x …
Alan's user avatar
  • 1,594
5 votes
1 answer
641 views

Theorems that tell if an explicit analytical solution is possible for nonlinear PDEs

Are there any theorems that tell if a particular nonlinear PDE can be solved explicitly by analytical methods? Where analytical methods I refer to methods such as power series or any methods that use …
Alan's user avatar
  • 1,594
1 vote
0 answers
101 views

Suggestion for books in Pertubation theory with an emphasis on the theory

As the title suggest I am looking for another good coverage of the theory of Pertubation theory. Currently I am working through Murodock's book: Pertubations: Theory and Methods. But I am rest assure …
Alan's user avatar
  • 1,594
1 vote
1 answer
2k views

Lipschitz functions and $W^{1,\infty}$

I am not sure my question is research type, but I am sure I can find here an answer. So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295: Theorem 4 (C …
Alan's user avatar
  • 1,594
7 votes
2 answers
562 views

Relativistic Control Theory

I am looking for literature that combines General relativity and control theory. So far I found a video lecture on "Integrability meets Control Theory: Harmonic maps in GR", other than that not so mu …
Alan's user avatar
  • 1,594
-2 votes
1 answer
220 views

Solving a nonlinear PDE numerically

I want to solve numerically the following PDE: $$ u_x + u_t - (u_{xt})^2 = u(x,t) $$ The boundary conditions are no concern of mine, pick the ones that work. So which numerical method will work for t …
Alan's user avatar
  • 1,594
0 votes
1 answer
247 views

Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of: $$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$ $$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ,u(0,t …
Alan's user avatar
  • 1,594
-4 votes
1 answer
571 views

Derivatives of infinite order [closed]

Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature? For example, can one make sense of $$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 \partia …
Alan's user avatar
  • 1,594
7 votes
0 answers
333 views

Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory? I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has P …
Alan's user avatar
  • 1,594
1 vote
0 answers
80 views

boundedness of a sequence $ \in L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$ implies that its tem... [closed]

Hi I have the next claim which I would like to find a proof of it. I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in L^\infty(I,H^1(M))\ …
Alan's user avatar
  • 1,594
6 votes
1 answer
387 views

Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrody...

Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics? Tried my luck with Google's search engine, didn't show much info. I guess you can try to use these features …
Alan's user avatar
  • 1,594